Explicit Formulas for the p-adic Valuations of Fibonomial Coefficients

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1 Journl of Integer Sequences, Vol , Article Exlicit oruls for the -dic Vlutions of ionoil Coefficients Phhinon Phunhy nd Prnong Pongsrii 1 Dertent of Mthetics culty of Science Silorn University Nhon Ptho 7000 Thilnd hhinon@gil.co rnong@gil.co Astrct We otin exlicit foruls for the -dic vlutions of ionoil coefficients which extend soe results in the literture. 1 Introduction The ioncci sequence n n 1 is given y the recurrence reltion n n 1 + n for n with the initil vlues 1 1. or ech 1 nd 1, the ionoil coefficients re defined y , 1 where n is the nth ioncci nuer. If 0, we define 1 nd if >, we define 0. It is well nown tht is n integer for ll ositive integers nd. So it 1 Prnong Pongsrii receives finncil suort jointly fro The Thilnd Reserch und nd culty of Science Silorn University, grnt nuer RSA Prnong Pongsrii is the corresonding uthor. 1

2 is nturl to consider the divisiility roerties nd the -dic vlution of. As usul, lwys denotes rie nd the -dic vlution or -dic order of ositive integer n, denoted y ν n, is the exonent of in the rie fctoriztion of n. In ddition, the order or the rn of ernce of n in the ioncci sequence, denoted y zn, is the sllest ositive integer such tht n. The ioncci sequence nd the tringle of ionoil coefficients re, resectively, A nd A in OEIS [5]. Also see A nd A0067 for signed ionoil tringle nd centrl ionoil coefficients, resectively. In 1989, Knuth nd Wilf [8] gve short descrition of the -dic vlution of C where C is regulrly divisile sequence. However, this does not give exlicit foruls for. Then recently, there hs een soe interest in exlicitly evluting the -dic vlution of ionoil coefficients of the for. or exle, Mrques nd Trojovsý [10, 11] nd Mrques, Sellers, nd Trojovsý [1] del with the cse + 1, 1. Bllot [, Theore 4.] extends the Kuer-lie theore of Knuth nd Wilf [8, Theore ], which gives the -dic vlution of ionoils, to ll Lucsnoils, nd, in rticulr, uses it to deterine exlicitly the -dic vlution of Lucsnoils of the for, for ll U nondegenerte fundentl Lucs sequences U nd ll integers > 0, [, Theore 7.1]. Note tht in the forul given y Mrques nd Trojovsý [11, Theore 1] for U nd + 1, only the cse of even is ctully exlicitly couted. It ers, using the theore of Bllot [1, Theore 7.1], tht their stted result for odd is correct only for ries for which does not divide z, where z is the rn of ernce of in the ioncci sequence. Also see Exles 16 nd 18 in this rticle. OururoseistoextendBllot stheore, Theore7.1, inthecseu nd > 0 nd otin exlicit foruls for l 1 l, where l 1 nd l re ritrry ositive integers such tht l 1 > l. This leds us to study the -dic vlutions of integers of the fors l l1 l! or!, 1 where ±1 od. or instnce, we otin in Exle 17 the following result: for ositive integers,,l with, nd rie distinct fro nd 5, if ±1 od 5, then { l +ν z +ν l, if z l; ν 0, otherwise. urtherore, if ± od 5, then l ν 0, if l 1 ε od z; +ν z +ν l, if l 0 od z;, if l 0,1 ε od z nd is even; 1 +ν z, if l 0,1 ε od z nd is odd,

3 where ε 1 if nd hve different rity nd ε 0 otherwise. We lso otin the corresonding results for {,5} in Exles 15 nd 19. These extend ll the in results in [10, 11, 1] nd Bllot s theore, Theore 7.1, in the cse U. Recll tht for ech x R, x is the lrgest integer less thn or equl to x, {x} is the frctionl rt of x given y {x} x x, nd x is the sllest integer lrger thn or equl to x. In ddition, we write od to denote the lest nonnegtive residue of odulo. We lso use the Iverson nottion: if P is theticl stteent, then { 1, if P holds; [P] 0, otherwise. or exle, [5 1 od 4] 0 nd [ 1 od 4] 1. We orgnize this rticle s follows. In Section, we give soe reliinries nd useful results which re needed in the roof of the in theores. In Section, we give exct foruls for the -dic vlutions of integers 1. In Section 4, we ly the results otined in Section to ionoil coefficients. Our ost generl theore is Theore 1. inlly, in Section 5, we give the -dic vlutions of soe secific su-filies of ionoil coefficients of tye 1, since generlly, the ore secific the fily, the shortest the forul ecoes. or ore infortion relted to ioncci nuers, we invite the reders to visit the second uthor s Reserchgte ccount [] which contins soe freely downlodle versions of his ulictions [5, 6, 7, 1, 14, 15, 16, 17, 18, 19, 0, 1, ]. Preliinries nd les Recll tht for ech odd rie nd Z, the Legendre syol is defined y Then we hve the following result. 0, if ; 1, if is qudrtic residue of ; 1, if is qudrtic nonresidue of. Le 1. Let 5 e rie nd let nd n e ositive integers. Then the following stteents hold. i If >, then 5 0 od. ii n if nd only if zn. iii z +1 if nd only if or od 5, nd z 1 otherwise. iv gcdz, 1.

4 Proof. These re well nown results. or exle, i nd ii cn e found in [4,. 410] nd [6], resectively. Then iii follows fro i nd ii. By iii, z ± 1. Since gcd,±1 1, we otin gcdz, 1. This roves iv. Lengyel s result nd Legendre s forul given in the following les re iortnt tools in evluting the -dic vlution of ionoil coefficients. We lso refer the reder to [10, 11, 1, 15] for other siilr lictions of Lengyel s result. Le. Lengyel [9] or n 1, we hve 0, if n 1, od ; ν n 1, if n od 6; ν n+, if n 0 od 6, ν 5 n ν 5 n, nd if is rie distinct fro nd 5, then { ν n+ν z, if n 0 od z : ν n 0, if n 0 od z, Le. Legendre s forul Let n e ositive integer nd let e rie. Then ν n! n 1 In the roof of the in results, we will del with lot of clcultion involving the floor function. So it is useful to recll the following results. Le 4. or n Z nd x R, the following holds i n+x n+x, ii {n+x} {x}, { 1, if x Z; iii x+ x 0, if x Z, { 1 {x}, if x Z; iv { x} 0, if x Z, { x+y, if {x}+{y} < 1; v x+y x+y+1, if {x}+{y} 1, vi x x n n for n 1. 4.

5 Proof. These re well-nown results nd cn e roved esily. or ore detils, see in [1, Exercise 1,. 7] or in [, Chter ]. We lso refer the reder to [14] for nice liction of v. The next le is used often in counting the nuer of ositive integers n x lying in residue clss od q, see for instnce in [4, Proof of Le.6]. Le 5. or x [1,,, q Z nd q 1, we hve x 1 q 1 n x n od q q. Proof. Relcing y +q nd lying Le 4, we see tht the vlue on the right-hnd side of is not chnged. Oviously, the left-hnd side is lso invrint when we relce y +q. So it is enough to consider only the cse 1 q. Since n od q, we write n +q where 0 nd +q x. So x q. Therefore 1 n x n od q 1 0 x q x x 1 +1 q q. q It is convenient to use the Iverson nottion nd to denote the lest nonnegtive residue of odulo y od. Therefore we will do so fro this oint on. Le 6. Let n nd e integers, ositive integer, r n od, nd s od. Then n n [r < s]. Proof. By Le 4i nd the fct tht 0 r <, we otin n n r + r n r r + n r. Siilrly, s. Therefore n is equl to n r s + r s n r s { r s n +, if r s; n 1, if r < s. 5

6 The -dic vlution of integers in secil fors In this section, we clculte the -dic vlution of l! nd other integers in siilr fors. Theore 7. Let e rie nd let 0, l 0, nd 1 e integers. Assue tht ±1 od nd let δ [l 0 od ]. Then l 1 l { l l 1 } +ν!, if 1 od ; l ν! 1 δ +ν l 1!, if 1 od nd is even; l 1 1δ { l l 1 } +ν!, if 1 od nd is odd. We rer tht if 1 or, then the exressions in ech cse of this theore re ll equl. Proof. The result is esily verified when 0 or l 0. So we ssue throughout tht 1 nd l 1. We lso use Les 4i, 4vi, nd 5 reetedly without reference. By Legendre s forul, we otin ν l! l j j1 l j + j1 j+1 ro, it is iedite tht for 1, we otin So we ssue throughout tht. ν l! l 1 1 l j +ν l!. Cse 1. 1 od. Then, for every 0, 1 od nd l l 1 + l l 1 l +. Therefore the su l j j1 ering in is equl to l j 1 + j1 l l j1 l j l +ν j1!. { } l l j l 1 l 1. Cse. 1 od. Then for 0, we hve 1 od if is even, nd 1 od if is odd. Therefore l l 1 + l l 1 l + if 0 nd is even, l l +1 l l +1 + l if 0 nd is odd. 6

7 Therefore the su l j j1 ering in is equl to 1 j j 0 od l 1 j l j 1 + j l j j od l By Le 4iii, we see tht l + l + l l l 1 j j 1 od + 1 l j j j 1 od l + l l + l l l [l 0 od ] δ. Therefore if is even, then 4 is equl to l 1 1 l l + l + l l + l l l 1 1 δ, nd if is odd, then 4 is equl to l l l + 1 l + l l l + l l + l { } l 1 1 l 1 δ. This coletes the roof.. 4 We cn coine every cse in Theore 7 into single for s given in the next corollry. Corollry 8. Assue tht,, l,, nd δ stisfy the se ssutions s in Theore 7. Then the -dic vlution of l! is l 1 { } l 1 δ [ 1 od ] { } l l +δ 1 [ 1 od ]+ν!. 5 7

8 Proof. This is erely cointion of ech cse fro Theore 7. or exle, when 1 od, the right-hnd side of 5 reduces to l 1 { } l l 1 δ [ 1 od ]+ν! l 1 δ +ν l 1!, if is even; l 1 { 1 1 δ l } l +ν!, if is odd, which is the se s Theore 7. The other cses re siilr. We leve the detils to the reder. l Next we del with the -dic vlution of n integer of the for 1 l! where,, l 1, l, nd re ositive integers. It is nturl to ssue l 1 l > 0. In ddition, if, then the ove exression is reduced to l1 l!, which cn e evluted y using Theore 7. We consider the cse in Theore 9 nd the other cse in Theore 10. Theore 9. Let e rie, let e nonnegtive integer, nd let,, l 1, l e ositive integers stisfying nd l 1 l > 0. Assue tht ±1 od. Then the following stteents hold. i If 1 od, then l1 l ν! l { } 1 l 1 l1 l 1 ii If 1 od nd od, then l1 l ν! l 1 l 1 1 iii If 1 od nd od, then l1 l ν! l 1 l 1 1 l1 l +ν!. { } l1 l [ 1 od ]!. [l 1 l od ]+ν l1 l { l } 1 +l [ 1 od ]!. [l 1 l od ]+ν l1 l We rer tht if 1, the exressions in ech cse of this theore re equl. 8

9 Proof. The result is esily checed when 0, nd s discussed ove, if, then the result cn e verified using Theore 7. So we ssue throughout tht 1 nd >. Siilr to the roof of Theore 7, we use Les 4i, 4vi, nd 5 reetedly without reference. Then, s for, we otin ν l1 l! l1 j l j + l1 j l j j1 j1 We see tht when 1, 6 ecoes ν l1 l! l 1 l 1 1 j+1 l1 j l j +ν l1 l +ν l1 l!.!. 6 So ssue throughout tht. We egin with the roof of i. Suose tht 1 od. or ech 1 j, we hve l1 j l j l1 j l 1 l j l + l 1 l l 1 j l j l 1 l + l1 l. Then the su l 1 j l j j1 ering in 6 is equl to l 1 j l j l1 l l1 l 1 j 1 j } 1 1 l 1 1 l 1 l 1 1 { l1 l 1 { } l1 l. l This roves i. So fro this oint on, we ssue tht 1 od. or ech 1 j, we hve l1 j l j l1 j 1 j l 1 l j 1 j l + 1 j l 1 1 j l l 1 j l j 1 j l 1 1 j l 1 j l 1 1 j l + l 1 j l j 1 j l 1 l + 1 j l 1 l, if od ; l 1 j l j 1 j l 1 +l + 1 j l 1 +l, if od. 9

10 Cse 1. od. Then the su l 1 j l j j1 ering in 6 is equl to l1 l 1 j l 1 l l 1 1 j j l 1 j l 1 l 1 1 Oserve tht j 1 j + 1 j 1 j l1 l 1 j + 1 j l 1 l 1 j 1 j So we hve l1 l 1 j 1 j 1 j 1 j { 0, if is even; 1, if is odd. l1 l [ 1 od ].. 7 It reins to clculte the lst ter in 7. If l 1 l od, then we otin y Le 4iii tht { 1 j l 1 l 0, if is even; l1 l 1 j, if is odd; l1 l [ 1 od ]. Siilrly, if l 1 l od, then we otin y Le 4iii tht 1 j l 1 l, l1 l 1 j 1 j l1 l if is even; 1, if is odd; [ 1 od ]. In ny cse, 1 j l 1 l l1 l [ 1 od ] [l 1 l od ]. Therefore 7 is equl to l 1 l 1 l1 l 1 [l 1 l od ] l 1 l 1 1 { l1 l [ 1 od ]+ l1 l [ 1 od ] } [ 1 od ] [l 1 l od ]. 10

11 This roves ii. Next we rove iii. Cse. od. Siilr to Cse 1, the su j1 equl to l 1 l 1 1 l 1 l j l 1 j l j ering in 6 is l1 +l 1 j + 1 j l 1 +l 1 j 1 j l1 +l + [ 1 od ]+ 1 j l 1 +l 1 j If l 1 l od, then we otin y Le 4iii tht { 1 j l 1 +l 0, if is even; l 1 +l, if is odd; 1 j l 1 +l [ 1 od ]. Siilrly, if l 1 l od, then we otin y Le 4iii tht { 1 j l 1 +l, if is even; l 1 +l 1, if is odd; 1 j l 1 +l In ny cse, 1 j Therefore 8 is equl to l 1 l 1 l1 +l + 1 [l 1 l od ] l 1 l 1 1 This coletes the roof. l 1 +l [ 1 od ].. 8 l 1+l [ 1 od ] [l1 l od ]. { l 1 +l [ 1 od ]+ l 1 +l [ 1 od ] } [ 1 od ] [l 1 l od ]. Next we relce the ssution in Theore 9 y <. The clcultion follows fro the se ide so we si the detils of the roof. Although we do not use it in this rticle, it y e useful for future reference. So we record it in the next theore. Theore 10. Let e rie, let e nonnegtive integer, nd let,, l 1, l e ositive integers stisfying < nd l 1 l > 0. Assue tht ±1 od. Then the following stteents hold. 11

12 i If 1 od, then l1 l ν! l 1 l 1 1 { l1 l l1 l }+ν!. ii If 1 od nd od, then l1 l ν! l { } 1 l 1 l1 l [ 1 od ] 1 l1 l [l 1 l od ]+ν!. iii If 1 od nd od, then l1 l ν! l { } 1 l 1 l1 +l [ 1 od ] 1 l1 l [l 1 l od ]+ν!. Proof. We egin y writing ν l 1 l! s l1 j l j + j1 j+1 The second su ove is ν s in Theore 9. We leve the detils to the reder. l1 j l j l 1 l!. The first su cn e evluted in the se wy When we ut ore restrictions on the rnge of l 1 nd l, the exression ν l 1 l ering in Theores 9 nd 10 cn e evluted further. Nevertheless, since we do not need it in our liction, we do not give the here. In the future, we ln to ut it in the second uthor s Reserchgte ccount. So the interested reder cn find it there. 4 The -dic vlutions of ionoil coefficients Recll tht the inoil coefficients is defined y {!, if 0 ;!! 0, if < 0 or >. 1.!

13 A clssicl result of Kuer sttes tht for 0, ν is equl to the nuer of crries when we dd nd in se. ro this, it is not difficult to show tht for ll ries nd ositive integers,, with, we hve ν, or ore generlly, ν ν. Knuth nd Wilf [8] lso otin the result nlogous to tht of Kuer for C-noil coefficient. However, our urose is to otin ν is n exlicit for. So we first exress ν in ters of the -dic vlution of soe inoil coefficients in Theore 11. Then we write it in for which is esy to use in Corollry 1. Then we ly it to otin the -dic vlution of ionoil coefficients of the for l 1 l. Theore 11. Let 0 e integers. Then the following stteents hold. i Let 6, 6, nd let r od 6 nd s od 6 e the lest nonnegtive residues of nd odulo 6, resectively. Then r+ r s+ s+ r s ν ν [r < s]ν. 6 ii ν 5 ν5. iii Suose tht is rie,, nd 5. Let z, z, nd let r od z, nd s od z e the lest nonnegtive residues of nd odulo z, resectively. Then +z ν ν +[r < s] ν +ν z. z Proof. We will use Les 4i nd 5 reetedly without reference. In ddition, it is useful to recll tht for every, N, ν ν +ν nd if, then ν ν ν. Since the foruls to rove clerly hold when 0 or, we ssue nd 1 <. 1

14 By Le, we otin, for every l 1, ν 1 l Then we otin fro the definition of ν 1 n l n od 6 1 n l n od 6 ν n n l n 0 od 6 1 n l n 0 od 6 ν n ν n+ l+ l + + ν 6j j l 6 l+ l + + ν j j l 6 l+ l l + +ν nd fro 9 tht!. 9 ν 1 ν 1 ν ν! ν! ν! The exression in the first renthesis in 10 is equl to r + r + r s + r s+ s + s r r+ r s r s+ + s s r + r s+ s Siilrly, the exression in the second renthesis is r r s s r s Therefore 10 ecoes r+ r s+ s+ r s ν x+y! +ν x!y! 11 14

15 where x nd y. By Le 4v, we see tht 6 6 x+y! x+y x!y! y, if {x}+{y} < 1; x+y y x+1, if {x}+{y} 1;, if {x}+{y} < 1; +6 6, if {x}+{y} 1. By Le 4ii, we otin { r s {x} + r s } { } { r s s nd {y} + s } s If r s, then {x}+{y} { } r s 6 + s r s+ s r < 1. If r < s, then we otin y Le iv tht {x}+{y} { } s r 6 + s s r 1 + s 1+ r 1. Therefore x+y! x!y!, if r s; , if r < s. Sustituting 1 in 11, we otin rt i of this theore. The clcultion in rts ii nd iii re siilr, so we give fewer detils thn given in rt i. By Le, for every l 1, we hve ν 5 1 l ν 5 n ν 5 n ν 5 l!, which ilies ν 5 1 n l 1 n l ν 5! ν 5! ν 5! ν 5 or iii, we ly Les nd 1iv to otin ν 1 l ν n 1 n l n 0 od z 1 l z ν l z ν z+ 1 n l n 0 od z l z ν z l! + ν z. z. ν n+ν z As in rt i, the ove ilies tht x+y! ν ν +x+y x yν z, 1 x!y! 15

16 where x nd y. In ddition, if r s, then {x}+{y} < 1 nd if r < s, then z z {x} + {y} 1. Therefore 1 cn e silified to the desired result. This coletes the roof. By Theore 11ii, we see tht the 5-dic vlutions of ionoil nd inoil coefficients re the se. So we focus our investigtion only on the -dic vlutions of ionoil coefficients when 5. Clculting r nd s in Theore 11i in every cse nd writing Theore 11iii in nother for, we otin the following corollry. Corollry 1. Let,, r, nd s e s in Theore 11. Let A ν! ν! ν!, nd for ech rie,5, let A ν! ν z! ν z!. Then the z following stteents hold. A, if r s nd r,s,1,,,4,; i ν ii or,5, we hve A +1, if r,s,1,,,4,; A +, if r < s nd r,s 0,,1,,,, 1,4,,4,,5; A +, if r,s 0,,1,,,,1,4,,4,,5. ν { A, if r s; A +ν z, if r < s. Proof. or i, we hve 0 r 5 nd 0 s 5, so we cn directly consider every cse nd reduce Theore 11i to the result in this corollry. In ddition, ii follows directly fro 1. In series of ers see [11] nd references therein, Mrques nd Trojovsý otin forul for ν only when +1. Then Bllot [] extends it to ny cse >. l1 Corollry 1 enles us to coute ν. We illustrte this in the next theore. l Theore 1. Let,, l 1, nd l e ositive integersnd. Let 5 e rie. Assue tht l 1 > l l nd let 1 l nd z. Then the following stteents z hold. 16

17 l1 i If od, then ν l is equl to ν, if l 1 l od or l 0 od ; ++ν +ν, if l 1 0 od nd l 0 od ; +1+ν +ν, if l 1 1 od nd l od ; +1 +ν, if l 1 od nd l 1 od, nd if od, then ν l1 l is equl to ν, if l 1 l od or l 0 od ; ++ν +ν, if l 1 0 od nd l 0 od ; +1 +ν, if l 1 1 od nd l 1 od ; +1+ν +ν, if l 1 od nd l od. ii Let 5 e n odd rie nd let r l 1 od z nd s l od z. If ±1 od 5, then l1 ν [r < s] +ν l +ν z +ν, l1 nd if ± od 5, then ν is equl to l +ν z +ν +ν, if l 1 0 od z nd l 0 od z; +ν, if r > s, l 1,l 0 od z, nd is even; +ν z +ν +ν, if r < s, l 1,l 0 od z, ν, +1 +ν +ν, 1 +ν z +ν, if r s or l 0 od z; nd is even; if r > s, l 1,l 0 od z, nd is odd; if r < s, l 1,l 0 od z, nd is odd. 17

18 Rer 14. In the roof of this theore, we lso show tht the condition r s in Theore 1ii is equivlent to l 1 l l [ od ] od z. It sees ore nturl to write r s in the stteent of the theore, ut it is ore convenient in the roof to use the condition l 1 l l [ od ] od z. l1 Proof of Theore 1. We ly Corollry 1 to clculte ν with l 1, l l, r l 1 od 6, nd s l od 6. or convenience, we lso let r l 1 od, nd s l od. Therefore A given in Corollry 1 is l1 1 l A ν! ν 1 l1! ν 1 l 1!. 14 By Corollry 8, the first ter on the right-hnd side of 14 is equl to { } l l1 l1 [l 1 0 od ] [ 0 od ]+ν l [r 0] r [ 0 od ]+ν Siilrly, the second ter is l [s 0] s [ 0 od ]+ν! l1!. 15 l!. 16 To evlute the third ter on the right-hnd side of 14, we divide the roof into two cses ccording to the rity of nd. Cse { 1. od. Oserve tht l 1 l od if nd only if r s. In ddition, l1 l } { r } s nd r s [r < s ]. Then y Theore 9, the third ter on the right-hnd side of 14 is equl to { } l 1 l 1 1 r s 1 [ 0 od ] [r s l1 l ]+ν l 1 l 1 1 r s 1 +[r < s ] [ 0 od ] [r s ]!. 17 l1 l +ν Recll tht otin y Le 6 tht l 1 nd l. Since is even, 1 od nd we l1 l [r < s ]. 18!

19 l Therefore ν 1 l! is equl to By Corollry 8, ν! is equl to ν! [r < s! ]ν ν! l 1 1. l1 [r 0]+ν!.! in 17 nd then sustitute 15, 16, nd 17 l We sustitute the vlue of ν 1 l in 14 to otin A. We see tht there re soe cncelltions. or instnce, r [ 0 od ] [ 0 od ] 0. nd! l ν ν! ν! Then we otin 1 1 A [r 0]+ [s 0]+[r < s ][ 0 od ] 1 + [r s ]+ [r 0]+[r < s ]ν +ν Next we divide the clcultion of A into 4 cses: Cse 1.1. l 1 l od or l 0 od, Cse 1.. l 1 0 od nd l 0 od, Cse 1.. l 1 1 od nd l od, Cse 1.4. l 1 od nd l 1 od.. 18 Since the clcultion in ech cse is siilr, we only show the detils in Cse 1.1 nd Cse 1.. So ssue tht l 1 l od. Then r s nd 18 ecoes 1 1 A [r 0]+ [r 0]+ [r 0]+ν Since , we see tht A ν. Next if l 0 od, then s 0 nd the se clcultion leds to A ν. Next ssue tht l 1 0 od nd l 0 od. Then r 0, s 0, nd 18 ecoes 1 1 A +[ 0 od ]+ 19 +ν +ν.

20 Oserving tht the su of the first three ters ove is equl to 1, we otin A 1+ν +ν. The other cses re siilr. Therefore A is ν, if l 1 l od or l 0 od ; 1+ν +ν, if l 1 0 od nd l 0 od ; +ν +ν, if l 1 1 od nd l od ; 1 +ν, if l 1 od nd l 1 od. Recll tht r l 1 od 6 nd s l od 6. Therefore 0, if l 1 0 od ; r, if is even nd l 1 od or if is odd nd l 1 1 od ; 4 if is even nd l 1 1 od or if is odd nd l 1 od, nd 0, if l 0 od ; s, if is even nd l od or if is odd nd l 1 od ; 4 if is even nd l 1 od or if is odd nd l od. l1 To otin the forul for ν l, we divide the clcultion into 4 cses: Cse 1.1 to Cse 1.4 s efore. Then we consider the vlues of r nd s in ech cse, nd sustitute A in Corollry 1. This leds to the desired result. Since the clcultion in ech cse is siilr, we only give the detils in Cse 1.. In this cse, A + ν + ν, r,s,4 if nd re odd, nd r,s 4, if nd re even. By Corollry 1, we otin { l1 A +, if nd re odd; ν l A +1, if nd re even, +1+ν +ν, s required. The other cses re siilr. Cse. od. The clcultion in this cse is siilr to Cse 1, so we oit soe detils. By Theore 9, the third ter on the right-hnd side of 14 is equl to } l 1 l 1 1 { r +s 1 [ 0 od ] [l 1 l od ] l1 l +ν!. 19 0

21 Since is odd, l 1 r od nd we otin y Le 6 tht l1 l B where B [r,s {0,1,0,,,}]. Siilr to Cse 1, ν l 1 l! ν! Bν ν.!! is Then we evlute ν! y Corollry 8, nd sustitute ll of these in 14 to otin tht A is equl to 1 1 [r 0] r 1 [ 0 od ]+ [s 0] } s + { + r +s 1 [ 0 od ]+. +Bν +ν [l 1 l od ]+ r Then we divide the clcultion into 4 cses nd otin tht A is ν, if l 1 l od or l 0 od ; 1+ν +ν, if l 1 0 od nd l 0 od ; 1 +ν, if l 1 1 od nd l 1 od ; +ν +ν, if l 1 od nd l od. We illustrte the clcultion of A ove only for the cse l 0 od since the other cses re siilr. So suose l 0 od. So s 0. If r 0, then it is esy to see tht A is equl to ν. So ssue tht r 0. Then A is equl to x + y + ν, where x 1 y r [ 0 od ] { 0, if is odd; } { r [ 0 od ]+ r 1, if is even, { 0, if is odd; 1, if is even. l1 ThereforeA ν, srequired. AsinCse1, wedividetheclcultionofν l into 4 cses ccording to the vlue of A, which leds to the desired result. This roves i. 1

22 or ii, we ly Corollry 1 with l 1 nd l. or convenience, we let r l 1 od z nd s l od z. The clcultion of this rt is siilr to tht of rt i, so we oit soe detils. We hve A ν l1 z! ν l z! ν l1 l z!. 0 Cse 1. ±1 od 5. Then y Le 1iii, 1 od z. By Corollry 8, the first ter on the right-hnd side of 0 is equl to l 1 1 z 1 +ν l1! z l 1 1 z 1 r z +ν nd siilrly, the second ter is [r 0] r [ 1 od ]+ z l1!, z l 1 z 1 s z +ν By Theore 9, the third ter is l 1 l 1 r s z 1 z +[r < s ] l!. z Since 1 od z, we otin y Le 6 tht l1 l [r < s ]. z l Therefore ν 1 l! is equl to z [r 0] 1 r z l1 l +ν!. z ν! [r < s! ]ν ν.! As usul, the first ter ove cn e evluted y Corollry 8 nd is equl to l 1 1 r z 1 z +ν l1!. z We sustitute ll of these in 0 to otin A [r < s ]+ν +ν.

23 Since 1 od z, r r nd s s. Sustituting A nd lying Corollry 1, we otin the desired result. Cse. ± od 5. Then y Le 1iii, 1 od z. By Corollry 8, the first ter on the right-hnd side of 0 is equl to l 1 1 z 1 [r 0] r z [ 1 od ]+ν l1!. z Siilrly, the second ter is l 1 z 1 [s 0] s z [ 1 od ]+ν l!. z or the third ter, we divide the roof into two cses ccording to the rity of nd. Cse.1. od. Then y Theore 9, the third ter on the right-hnd side of 0 is equl to l 1 l 1 r s z 1 z +[r < s ] [ 1 od ] [r s ] l1 l +ν!. z As in Cse 1, we ly Le 6 to write l1 l [r < s ], z l nd then use Corollry 8 to show tht ν 1 l! is equl to z l 1 1 z 1 l1 [r 0]+ν! z [r < s ]ν ν!.! Sustituting ll of these in 0, we see tht A is equl to [r 0]+ [s 0]+[r < s ][ 1 od ]+ [r s ]+ [r 0] +[r < s ]ν +ν ν, if l 1 l od z or l 0 od z; +ν +ν, if l 1 0 od z nd l 0 od z; +ν, if l 1,l 0 od z nd r > s ; +ν +ν, if l 1,l 0 od z nd r < s.

24 Recll tht r l 1 od z nd s l od z. If nd re even, then 1 od z, r r, nd s s l1, nd we cn otin ν y sustituting A in l Corollry 1. Suose nd re odd. Then r r od z nd s s od z nd thus when r nd s re oth nonzero or re oth zero, we hve r s if nd only if r s. l1 Siilr to the ove, we cn otin ν y the sustitution of A in Corollry 1. l1 We see tht ν l is equl to ν, l if l 1 l od z or l 0 od z; +ν z +ν +ν, if l 1 0 od z nd l 0 od z; +ν, if r > s, l 1,l 0 od z, nd is even; +ν z +ν +ν, if r < s, l 1,l 0 od z, nd is even; +1 +ν +ν, if r > s, l 1,l 0 od z, nd is odd; 1 +ν z +ν, if r < s, l 1,l 0 od z, nd is odd. Since od, we see tht od z nd therefore l 1 l od z r s 1 So the condition l 1 l od z cn e relced y r s. Cse.. od. The clcultion in this cse is siilr to tht given efore. So we si soe detils. By Theore 9, the third ter on the right-hnd side of 0 is equl to l 1 l 1 l1 [ 1od]B 1 [l 1 l odz]+ν z 1 l!, z { } where B 1 r +s r +s z z +[r +s > 0]+[r +s > z]. Since 1 od z, we otin y Le 6 nd strightforwrd verifiction tht l1 l ε, z where ε [ r od z < s ] [r 0 nd s 0]+[r +s > z]. Then y Corollry 8, l ν 1 l! is equl to z l 1 1 z 1 [r 0] r z +ν l1! B ν z 4!!,

25 where B εν. Since od, [ 1 od ] 1 [ 1 od ] nd + +[ 1 od ]. We sustitute ll of these in 0 to otin tht A is equl to [r 0]+ [s 0]+[l 1 l od z] +[r +s > 0]+[r +s > z][ 1 od ]+B +ν ν, if l 1 l od z or l 0 od z; +ν +ν, if l 1 0 od z nd l 0 od z; +ν, if l 1,l 0 od z nd r +s < z; +ν +ν, if l 1,l 0 od z nd r +s > z. Recll tht r l 1 od z nd s l od z. Suose tht is odd nd is even. Then r r nd s s od z. Moreover, if s 0, then s z s nd thus r < s r +s < z nd r > s r +s > z. Siilrly, if is even nd is odd, then r r od z nd s s, nd for r 0, we hve r < s r +s > z nd r > s r +s < z. l1 rotheoveoservtionndthesustitutionofa incorollry1,weseethtν l is equl to ν, +ν z +ν +ν +1 +ν +ν, if l 1 l od z or l 0 od z;, if l 1 0 od z nd l 0 od z; if r > s, l 1,l 0 od z, nd is odd; 1 +ν z +ν, if r < s, l 1,l 0 od z, nd is odd; +ν, if r > s, l 1,l 0 od z, nd is even; +ν z +ν +ν, if r < s, l 1,l 0 od z, nd is even. Since od, we see tht od z nd therefore Coining this with 1, we conclude tht This coletes the roof. l 1 l od z r s. l 1 l l [ od ] od z r s. 5

26 5 Exles In this lst section, we give severl exles to show lictions of our in results. We lso recll fro Rer 14 tht the condition r s in Theore 1ii cn e relced y l 1 l l [ od ] od z. In the clcultion given in this section, we will use this oservtion without further reference. Exle 15. Let,,ndleositiveintegersnd. Wessertthtforl 0od, we hve l +1 ν ε 1 ε +ε 1ε, where ε 1 [l od ], ε [ od ], ε 1 [l 1 od ], nd ε [ od ]. In ddition, if l 0 od, then l ν ++ν l. l Proof. We ly Theore 1 to verify our ssertion. Here nd 1 0. So we ieditely otin the following: if od, then l 0, if l 1 od ; ν ++ν, if l 0 od ;, if l od, nd if od, then l ν +1 0, if l 1 od ; ++ν, if l 0 od ;, if l 1 od. +1 This roves. If l 0 od, then l nd ν is equl to which ilies. ν ν l+ν ν +ν l, Exle 16. Sustituting l 1 in Exle 15, we see tht +1 ν [ od ] { 0, if od ; 4, if od. +1 Our exle lso ilies tht 4 still holds for the -dic vlutions of +c 5 +1, 1, etc. 6, 7,

27 Exle 17. Let,, nd l e ositive integers,, nd rie distinct fro nd 5. If ±1 od 5, then l ν +ν z +ν l [l 0 od z], nd if ± od 5, then 0, if l 1 ε od z; l +ν z +ν l, if l 0 od z; ν, if l 0,1 ε od z nd is even; 1 +ν z, if l 0,1 ε od z nd is odd, where ε [ od ]. l Proof. Siilr to Exle 15, we verify this y lying Theore 1. Here, z 0, r l od z, nd s od z. We first ssue tht 1 z ±1 od 5. Then y Le 1, we hve 1 od z. Therefore s 1, r l od z, nd l ν +ν +ν z [l 0 od z]. Siilrly, if ± od 5 nd od, then we otin y Le 1 nd Theore 1 tht 0, if l 1 od z; l +ν +ν z, if l 0 od z; ν, if l od 0,1z nd is even; 1 +ν z, if l 0,1 od z nd is odd. In ddition, if ± od 5 nd od, then 0, if l 1 od z; l +ν +ν z, if l 0 od z; ν, if l 0, 1 od z nd is even; 1 +ν z, if l 0, 1 od z nd is odd. It reins to clculte ν when l 0 od z. In this cse, we hve l ν ν ν l+ν ν z +ν l. z This ilies the desired result. 7

28 Exle 18. Sustituting l 1 in Exle 17, we see tht for,5, we hve 0, if ±1 od 5 or od ; ν, if ± od 5, od, nd is even; 1 +ν z, if ± od 5, od, nd is odd. 5 Our exle lso ilies tht 5 still holds for the -dic vlutions of +c nd z+1. Siilrly, for,5, we hve 0, if ±1 od 5; ν, if ± od 5 nd is even; 1 +ν z, if ± od 5 nd is odd. In ddition, 6 lso holds when,5. urtherore, relcing is relced y l y z 1 6 for l 0,±1 od z nd, the forul ecoes 0, if ±1 od 5 or od ;, if ± od 5, od, nd is even; 1 +ν z, if ± od 5, od, nd is odd. Exle 19. We now tht the 5-dic vlutions of ionoil coefficients re the se s those of inoil coefficients. or exle, y Theore 11ii nd Kuer s theore, we otin l 5 l 5 ν 5 ν 5 5 +ν 5 5 l, for every,,l N with. Siilrly, ν 5 5 ν 5 l for every,,l N such tht 5 > l 5. l 5 Exle 0. Let,, nd l e ositive integers nd > l. Let l. Then ν l ν + + +ν ε 1 ε + +1 nd ε ε 4, 7 where ε 1 [ od ], ε [l od ], ε [ od ], nd ε 4 [l 1 od ]. Proof. Siilr to Exle 15, this follows fro the liction of Theore 1. So we leve the detils to the reder. 8

29 Exle 1. Let. We oserve tht { 1, if is even; 1 1, if is odd, which ilies, ν [ 1 od ]. 8 By siilr reson, we lso see tht for, ν 1 [ 0 od ]. 9 ro 7, 8, nd 9, we otin the following results: i if, then ν [ od ], ii if, then ν 5 is equl to + [ od ]+ +[ od ] [ od ] [ od ], iii if, then ν iv if 4, then ν 6 7 [ od ], [ od ]+ +1 [ od ]. Exle. Let 5 en odd rie nd let,, nd l e ositive integers, > l,, nd. Then the following stteents hold. z l z i If ±1 od 5, then ν l ii If ± od 5, then ν +ν +ν z [l 0,1 od z]+ν l is equl to, ν +ε 1 ε ε 5 +ν +ε ν z +ε 1 ε 4 1 ε 5 +ε ν z 0 where ε 1 [l 0 od z], ε [l 1 od z], ε [ 0 od ], ε 4 [l 1 od z], nd ε 5 [ od ]. 9

30 Proof. Siilr to Exle 17, this follows fro the liction of Le 1 nd Theore 1. Since i is esily verified, we only give the roof of ii. The clcultion is done in two cses. If ± od 5 nd od, then ν is equl to l ν, if l 0,1 od z; +ν z +ν +ν, if l 0,1 od z nd is even; +1 +ν +ν, if l 0,1 od z nd is odd, ν +ε 1 ε +ν +ε ν z, where ε 1 [l 0 od z], ε [l 1 od z], nd ε [ 0 od ]. If ± od 5 nd od, then ν, if l 0, 1 od z; ν l +ν, if l 0, 1 od z nd is even; 1 +ν z +ν, if l 0, 1 od z nd is odd, ν +ε 1 ε 4 +ε ν z, where ε 1,ε,ε re s ove nd ε 4 [l 1 od z]. Let ε 5 [ od ]. Then oth cses cn e coined to otin ii. Exle. Let. We oserve tht z7 8 nd { 7 7 1, if is even; , if is odd. 8 Therefore 7 7 ν 7 [ 1 od ] nd ν ro 0 nd 1, we otin the following results: i if, then ν 7 7 [ od ], ii if, then ν [ 0 od ] [ od ], iii if, then ν 7 7 is equl to 15 7 [ od ]+ +[ 0 od ] [ od ]. To ee this rticle not too lengthy, we ln to give ore lictions of our in results in the next rticle. 0

31 6 Acnowledgents We re very grteful to the nonyous referee for his/her creful reding, ind words of rise, nd ny vlule suggestions which gretly irove the resenttion of this rticle. Phhinon Phunhy receives scholrshi fro Science Achieveent Scholrshi of ThilndSAST. Prnong Pongsrii receives finncil suort jointly fro The Thilnd Reserch und nd culty of Science Silorn University, grnt nuer RSA References [1] T. M. Aostol, Introduction to Anlytic Nuer Theory, Sringer, [] C. Bllot, Divisiility of ionoils nd Lucsnoils vi generl Kuer rule, ioncci Qurt , [] R. L. Grh, D. E. Knuth, nd O. Ptshni, Concrete Mthetics : A oundtion for Couter Science, Second Edition, Addison Wesley, [4] T. Koshy, ioncci nd Lucs Nuers with Alictions, Wiley, 001. [5] N. Khochi nd P. Pongsrii, The eriod odulo roduct of consecutive ioncci nuers, Int. J. Pure Al. Mth , [6] N. Khochi nd P. Pongsrii, The generl cse on the order of ernce of roduct of consecutive Lucs nuers, Act Mth. Univ. Coenin., to er. [7] N. Khochi nd P. Pongsrii, The order of ernce of roduct of ioncci nuers, Contri. Discrete Mth., to er. [8] D. Knuth nd H. Wilf, The ower of rie tht divides generlized inoil coefficient, J. Reine Angew. Mth , [9] T. Lengyel, The order of the ioncci nd Lucs nuers, ioncci Qurt. 1995, 4 9. [10] D. Mrques nd P. Trojovsý, On divisiility roerties of ionoil coefficients y, J. Integer Sequences 15 01, Article [11] D. Mrques nd P. Trojovsý, The -dic order of soe ionoil coefficients, J. Integer Sequences , Article [1] D. Mrques, J. Sellers, nd P. Trojovsý, On divisiility roerties of certin ionoil coefficients y, ioncci Qurt , [1] K. Onheng nd P. Pongsrii, Susequences nd divisiility y owers of the ioncci nuers, ioncci Qurt ,

32 [14] K. Onheng nd P. Pongsrii, Jcosthl nd Jcosthl-Lucs nuers nd sus introduced y Jcosthl nd Tvererg, J. Integer Sequences 0 017, Article [15] P. Pongsrii, Exct divisiility y owers of the ioncci nd Lucs nuer, J. Integer Sequences , Article [16] P. Pongsrii, A colete forul for the order of ernce of owers of Lucs nuers, Coun. Koren Mth. Soc , [17] P. Pongsrii, ctoriztion of ioncci nuers into roducts of Lucs nuers nd relted results, JP J. Alger Nuer Theory Al , 6 7. [18] P. Pongsrii, Locl ehviors of the nuer of reltively rie sets, Int. J. Nuer Theory 1 016, [19] P. Pongsrii, ioncci nd Lucs nuers ssocited with Brocrd-Rnujn eqution, Coun. Koren Mth. Soc. 017, [0] P. Pongsrii, ioncci nd Lucs Nuers which re one wy fro their roducts, ioncci Qurt , [1] P. Pongsrii, Integrl vlues of the generting functions of ioncci nd Lucs nuers, College Mth. J , [] P. Pongsrii, ioncci nd Lucs nuers ssocited with Brocrd-Rnujn eqution II, Mth. Re., to er. [] P. Pongsrii, Reserchgte roject: ioncci nuers nd the order of ernce, htts://tinyurl.co/yrzog, 017. [4] P. Pongsrii nd R. C. Vughn, The divisor function on residue clsses I, Act Arith , [5] N. J. A. Slone, The On-Line Encycloedi of Integer Sequences, htts://oeis.org. [6] D. D. Wll, ioncci series odulo, Aer. Mth. Monthly , Mthetics Suject Clssifiction: Priry 11B9; Secondry 11B65. Keywords: ioncci nuer, inoil coefficient, ionoil coefficient, -dic vlution, -dic order, divisiility. Concerned with sequences A000045, A0067, A010048, nd A

33 Received Octoer 1 017; revised version received Mrch Pulished in Journl of Integer Sequences, Mrch Return to Journl of Integer Sequences hoe ge.