[a-g[;:;:ur].-«'<* '**-* For Ar = 2, these are the binomial coefficients and when dealing with these we shall use the usual notation:

Transcription

1 SUMS OF PARTTO STS GRALZD PASCAL TRAGLS V..HOGG ATT, R. Sa ose ate Uversty, Sao ose, Calfora 992 ad G. L.ALXADRSO Uversty of Sata Clara, Sata Clara, Calfora 93 the expaso clearly ad (k-d (xx 2.~ x k - f = Y* [ / " ] * ' k>2, > 0, =0 [ o ] * " U-/*»]*= [a-g[;:;:ur].-«'<* '**-* For Ar = 2, these are the bomal effcets ad whe dealg wth these we shall use the usual otato: ],-(' The problem of calculatg sums of the followg type for k = 2 was fr treated by urot [2] ad Ramus [] ad Ramus' method s outled [ ] : where j=0 = j- f t -y,_ f - f [ ] deotg the greate teger fucto. We wsh here to vegate for certa fxed k ad q the dfferet values of these sums as r rages from 0 to q - ad, further, the d f ereces betwee the sums, Let be a prmtve q th rt of uty the. TH MTHOD OF RAVUS k The 2TT - 2TT to = s * / s Q Q t ' ( t t o ) * ( ) ( 2 ) * ( 3 ) -*( ) 7

2 8 SUMS OF PARTTO STS GRALZD PASCAL TRAGLS [Aprl ( u*-}. / ) ( ", j a>*" f ( ) c o 2 ( ) c o ' V.. * ( ) a > " ". Multplyg each successve row by, " 7 ", a)" 2 '',, u>~' q ~, Q<r<q-, ad addg the products we get * [ ( ) * (, % ) - (,*%) * '] - tf *««/V«- ( ** <> **fur**** $=0 0=O q- / y.\-* gfr-2r; Qq-l - /. \ r - [ ( c2s o s e l) " w 2. c o s fjr \ \ s ffer / l r f=0 Sce the left sde s real, the effcet of / o the rght mu be zero, hece S(. 0=0 ' o( - 2rh Applyg the same techque to the expaso (x x 2 - x k ~ ) S(,k,q,r) = f " " f *- 0=0 q- &=o kzl 2 2 y efk-zjthr M k/2 M o(k - 2j fr Q q -* o=o s s 2. THCASSA-2, q = 3, oe fdhat,20 o(k - - 2r)r Q o(k - - 2rh Q for k odd for k eve Ths case reated [] ad more recetly [6]. From the formulas above oe easly showhat (k-dzf-rs. [ > f(-) s 2j f~\= [ >. s f] (s)*(o*(0 ---j[e»] By examg the table for s (r)/3 oe seehat the three dffereces [ S._ S />LL], [ c o $ fezt_ c o sk. a d - c o s k r are 0,, -. Ths problem appeared the Amerca Mathematcal Mothly May, 938 as Problem 3 (soluto by mma Lehmer) ad aga February, 96, as Problem 72. slghtly altered form t had appeared the Mothly 932 as Problem 397 (soluto by Morga Ward). t appeared as Problem B-6 the 97 Wllam Lowell Putam te. s

3 976] SUMS OF PARTTO STS GRALZD PASCAL TRAGLS 9 The case of four sums (q = ) yelds each case oly two dfferet or three dfferet sums, depedg o whether s odd or eve ad the dffereces the values are successve powers of 2, ahe reader ca verfy. Ths appeared Mathematcs Magaze vember-december, 92 as Problem 77 (soluto by. P. arke). 3. TH CAS* = 2*7 = Ths case wareated tersely the soluto to a problem posed by. P. arke the March, 939, ssue of the atoal Mathematcal Magaze, where the dffereces were observed to be smple ad predctable but the sums themselves were ot see to be reducble to smple form. We shall, therefore, treat ths very tereg case at legth, alog wth geeralzatos. sderato of the followg two fgures yelds values of x ad y: x - (a) Fgure 7 /,-.. x- y -y (b) = _ 7-V where the sgs are chose so thatx,/ are postve. We ote a he golde rato ad the a,j3are those of the Bet formulas for elemets of Fboacc ad Lucas sequeces,.e., f Fj =, F 2 = F = F-j F-2, > 2, ad Z. / = 7, L 2 = 3, L = L L. 2, > 2, the ad L - optf, where F ad L are the /7 f/7 Fboacc ad Lucas umbers, respectvely [3]. From Fg. a, oe seehat s = <k ad 2r _ a- s 2 ad from a P = oe cludehat s 2 From these oe ca ruct a table of values for s (m)/. The = l [ > a c o ( -/ r ' H r s 2( "- 2r)7r (-a) s *ks /3"s fc&] = l\ 2" 2a" s ( -~ 2<-$) s H-ldl ] for r = 0,,2, 3,.

4 20 SUMS OF PARTTO STS GRALZD PASCAL TRAGLS [Aprl Let us exame, for example, S(0m, 2,, 0): where L om s a Lucas umber. For/7 = 0m, to L[2 0m 2a 0m 2$ 0m ] = D O -[2 0m 2L 0m ] S<0m,Z,0)= -2 0m 2a 0m (a/2) - 2p 0m ($/2)l = -[2 0m a 0m2 $ 0m2 ] D 3 _ / r 9 0m,, / We ca tue to reduce these sumo the form /[2 A], where A s a Lucas umber or twce a Lucas umber ad ca, fact, form the followg table for the values of A ' Table 0m Wm 0m2 0m3 0m 0m 0m6 0m 7 0m8 0m9 r = 0 r = 2 r= 2LOm L Wm2 LtOm ~ L 0m2 ~ L 0m -2L jom -l-om7 ~ L 0m6 l~0m7 L-WmW LlOm- L 0m2 2 0m2 L Wm L 0m3 --Om -l-0m7-2l Wm7 ~L0m9 -LOm8 -Wm -Lom L-Wm L 0m 2L om L Wm6 L 0m --Wm6 -Ll0m9-2L om9 -Lwm -2LQm -l~0m3 -Lwm2 L 0m3 L 0m6 2L Wm6 L Om8 L Wm7 -LlOm8 Lwm- -Lwm -t-0m3-2l Wm3 -L-0m ~Ll0m L Wm L Wm8 2L Om8 Lwm0 Thus we have formulas for all sums of the form ad sce 2[ r t j ' r = 0,, 2, 3,, t=0 tl)-* - =0 we ote that the sum of the fve elemets o ay row of the above table mu be zero ad, furthermore, t s clear from the method of geeratg Pascal's Tragle that each elemet of Table mu be the sum of the elemet above t ad to the left of that. The followg he table of hgh ad low values of the elemets Table : H 2Lwm L Wm2 2L Wm2 L Wm 2L Wm L Wm6 2L Wm6 L Wm8 2L Wm8 LWm0 Table 2 L -L-Wm -2Lf0m -Lwm3-2L Wm3 -l-wm ~2L Wm -t~wm7-2lom7 -t~wm9 ~2L Wm9

5 976] SUMS OF PARTTO STS GRALZD PASCAL TRAGLS 2 The dffereces betwee the hghe value of the sums for gve ad the lowe value s, therefore, always of the form That (2L L )/. (2L L )/ = F s proved easly by ducto. We ote that for each there are oly three dfferet values for the fve sums ad that dffereces betwee the hgh ad low values are Fboacc umbers. Furthermore, the dffereces betwee the hgh ad mddle values, the mddle ad low values are aga Fboacc umbers. fact, the three Fboacc umbers have secutve subscrpts.. TH CAS k = 3 f q = ths case we are dealg wth fve sums of tromal effcets, ad, for r= 0,, 2,3,, But sce ad 2( - r)9.t M - [ ; ] / [ ] / [ r / 0 ] /... - l [ o s f, ] " o o s 2 s 2 -- e=o 2$Q = 2 = -a = P= 2QO*QZ, ( -! ) = 2 (- y = -p = a= 2 s &- S(,3,,r) = \ [ > a " s & 0» s f * 0 " s fc a " s dlzls."] b = [ 3" 2a s llohl 2$" C0S=jjL. These sums reduce each case to the form /[3 B], where B s foud Table 3: Tor 0m 0m 2 0m3 0m 0m 0m 0m 7 0m 8 0m 9 r = 0 2L0m LlOm ~ L Wm3 -t-0m L Wm3 2L Om L 0m -L-jOmg ~ L Wm9 L 0m8 Table 3 r = r = 2 LtOm- - L 0m 2Lwm -Wm Lwm 2L<o m 2 -l~wm Lo m 2 -t-om -LOm L>Wm -LOm6 2L Wm6 L 0m L Wm6 2L Om7 -LOm9 -LOm0 L-om7 -Lwm0 r = 3 -LlOm -LlQm2 l-0m 2L Om3 L Wm3 -LOm6 ~Wm7 l~wm6 2L Wm8 L Wm8 r = -Wm- -Lwm2 -l~wm3 L Wm2 2L mm L Wm -Lwm7 -Lwm8 f-wm7 2L Wm9 Aga, dffereces of the sums are Fboacc umbers. f oe exames cases for larger values of k ad usehe fact that, for oe seehat the sums wll be expressble the form q =, GO o) 2 <0 3 GO = 0, where C s a Lucas umber or twce a Lucas umber, ad the dffereces wll be secutve Fboacc umbers, the cases where k = 2,3 (mod ). other cases, the sumake o a at value or take o two values whch dffer by.

6 22 SUMS OF PARTTO STS GRALZD PASCAL TRAGLS [Aprl Here. TH CAS QFk = 2, q = 6 S(,2,6,r) = 2 s f ] "s zmm = 2"2(s/3) s ( - -f 2 s 2< ~ 2r) «] r=q,, -,, ad the sumake the form [2 D], where, for/"= 0, for example, D ca be foud Table. o Table 2m 2m 2m2 2m3 2m 2m 2m6 2m 7 2m8 2m9 2m 0 2m 2.3 6m 2 (ths breakd dow form = 0) 36m j6m 3 6m3 O 0673, O -36* 6m _ m _ * 36m6 The other sums, for r -,2,3,, ca be mputed easly ad, ot surprsgly, the large ad smalle sums dffer by a power of 3 or twce a power of 3. The Pell umbers/ are defed by the followg: 6. TH CAS OF k = 2, q = 8 Pf = P2 = 2, 2P - P - 2, > 2, ad we shall defe the Pell-Lucas sequece Q as satsfyg the same recurso relato but Q x =2, Q 2 =6. The rts of the auxlary equato x 2-2x - = 0 are, ths case, ad the Bet-type formulas ths case are, aalogously, y = y/2 ad 8 = - 2 P = L ad Q = T " S ' 7- For q = 8, the sums S(,2,8,r) for r = 0,, 2,, 7 ca be wrtte S(,2,8,r> - ( ; ) * ( / ) (, / )... - (,s f ) "s f h _- >, ( 2 s. c o s L /, c o $ j " c o s l / 2 c o s [ j " C Q S fcjw* A 2 s f j \*!!LzM!L 2s f ) "s L z T, /, c o s Z* " c o s TOLZM*' = r / V c o s (~2rh 2-2 /2 s 2( ~ 2r, Z 2-2 / (-8) /2 3<-2rhto L o 8 8

7 976] SUMS OF PARTTO STS GRALZD PASCAL TRAGLS v "" <M ca h CSj * * - H " t t o. «- < - t. - t. - " *~. t, M- l 2.. *»» - CSS 3 f - % - *> eg " - <* * - " «* - - ' c. * e c t O : F - fr 3 s- f s - «>! / S 0 0 s - u* u> / - / 3" s- r>, O j LO h M- C - * - Q3 - cp. h t >» «- s # - S to - S S - f M- S h! : L\

8 2 SUMS OF PARTTO STS GRALZD PASCAL TRAGLS [Aprl rss s*» s c S* CSj X «. 0 : c 0 c S- s 0 u> 0 o * ** h v- *» *««. < < S C S krs * < j s < (M s t, 0 u> u p c P * up * UP h* Sh. < *«. < 9 S - # LO H us «$ c«0b * u S x. t o o> S * l*> S, > C=) * ««. CD s

9 976] SUMS OF PARTlTf O\f STS GRALZD PASCAL TRAGLS 2 Oe ca reduce these sumo the form -[2 ], where s foud Table. S(,6,8,r) s smlar. o Dffereces betwee the large ad smalle sums are, ths case, powers of 2 tmes Pell or Pell-Lucas umbers. Further cases yeld more dffereces whch satsfy creasgly mplcated lear recurso relatos or mbatos of such relatos. Some of these, alog wth other techques for hadlg such problems wll appear a later paper. Some geeralzatoo multomal effcets appear []. RFRCS. L. Carltz, "Some Sums of Multomal effcets,!' to appear, The Fboacc Quarterly. 2. A. A. urot, Bull. ScL Math., Feb., V.. Hoggatt, r., Fboacc ad Lucas umbers, Houghto-Mffl, etto, Lehrbuch der mbetork, 2d ed., Teuber, Berl, Chr. Ramus, "Soluto gee'rale du probleme d'aalyse mbatore,". Ree Agew. Math., (83), pp A. M. ad. M. Yaglom, Challegg Mathematcal Problems wth lemetary Solutos, Vol., Holder Day, Sa Fracs, 96.