Sum of cubes: Old proofs suggest ew aalogues Joha Cigler Faultät für Mathemati, Uiversität Wie ohacigler@uivieacat Abstract We show how old proofs of the sum of cubes suggest ew aalogues 1 Itroductio I [1], [5] ad [7] some aalogues of the well-ow formula 1 1 (1) have bee foud I this ote I propose aother oe which is ispired by a old result of C Wheatstoe [6] He observed that the odd umbers ca be grouped i such a way that the idetities hold, which implies that Sice 11 5 7911 115 17 19 4 1 1 1 1 () 1 ( 1) () we get (1) Idetity (1) ca also be writte as 1 1, (4) because 1 1 (5) 1
A simple proof without words of (4) has bee give i [4], which I reproduce here: The simplest computatioal proof of (1) uses the trivial idetity gives the telescopig sum 1 which 1 1 1 1 1 1 aalogues 1 1 [ ] [ 1] [ 1] As usual we let [ ] 1 ad 1 [1] [] [ ] 0 1 It is clear that lim 1 ad lim 1 for
KC Garrett ad K Hummel [1] derived a combiatorial proof of the aalogue SO Waraar [5] proposed the idetity [ 1] [ 1] 1, [] 1 (6) 1 1 [ ] [ ] 1 (7) ad G Zhao ad H Feg [7] gave a combiatorial iterpretatio of 1 4( ) 1 1 [ ] 0 1 (8) For a computatioal proof observe that the left-had sums become telescopig by usig the idetities 1 1 1 1 a 1 a 1a 1 1 1 1 1 a 1a 1 a 1a 1 1 for a 0,1, A similar idetity is 1, (9) which follows from 1 1 1 1 1 1 1 1 1 This idetity gives the telescopig sum
1 1 1 1 1 [ ] [ ] 1 1 1 1 1 1 1 1 1 1 1 1 Thus we have obtaied our mai result Theorem 1 1 1 1 [ ] [ ] 1 (10) We ow give two further proofs which geeralize the beautiful proofs which we have setched i the itroductio ad which origially led to this aalogue We start with the followig well-ow (cf []) aalogue of () A telescopig proof uses the fact that [1] [ ] (11) 1 1 1 ( 1) 1 1 1 [ ] 1 [1] But formula (11) has also a ice combiatorial iterpretatio 0, Cosider the suares S i, ad associate with each poit i wi 1 (, ) i The weight of S is i 1 1 i 1 i0 0 w S [ ], S the weight (1) 4
The suare S is the uio of the hoos h 1, 0, 1,1,, 1, 1,, 1,, 0, 1, 1 The weight of the hoo h is 0 1 wh 1 1 The poit,0 will be called the base-poit of the suare As a example cosider the case 6 i matrix otatio 1 5 6 7 8 9 10 4 5 6 7 8 9 4 5 6 7 8 4 5 6 7 4 5 6 4 5 (1) Here we have wh 5 ( 1), wh 4 ( ) [], wh ( ) [5], Secod proof of Theorem 1 We first observe that formulae () have the ice aalogues ad more geerally This follows from the idetity 1 [1] [1 ] [] [5] [] [ ] [7] [9] [11] [] [ ] 1 1 [ ] [ ] (14) 1 1 1 1 1 1 1 1 1 1 1 1 0 0 [ ] [ ] 1 [ ] [ ] Usig (11) we get the desired result 5
1 1 1 1 1 1 [ ] [ ] 1 1 1 1 1 1 1 1 1 1 [1] 1 Third proof of Theorem 1 A combiatorial proof ca also be give alog the lies of the above proof without words For odd the uio R of the hoos h,, h 1 1 1 1 m is the uio of suares of sidelegth whose base poits have weight The weight of these suares is, 0 m 1 1 1 1 1 1 ( 1) 1 1 [ ] 1 [ ] [ ] 1 1 [ ] For the uio R of the hoos h,, h 1 1 1 1 m 1 1 is the uio of 1 suares whose base-poits have weights, 0 m, ad of two rectagles with side legths ad as i the blue regio i the above figure The weight of the uppermost rectagle is rectagle is 1 1 [ ] [ ] 1 [ ] [ ] ad the weight of the leftmost Thus the total weight of this regio is 6
1 1 1 1 1 m [ ] [ ] [ ] [ ] m0 1 1 m 1 4 m0 1 [ ] [ ] 1 1 1 4 1 1 4 1 [ ] [ ] 1 [ ] 1 1 [ ] 4 Thus we get w R 1 ws 1 1 ad thus agai (10) 1 1 1 1 4 Related results Let us ote some related results For ay seuece of positive itegers a () the sums ad 1 ai () i1 [ a( )] a( i) 1 i1 (15) ai () i1 [ a( )] ai ( ) 1 i1 (16) are aalogues of i1 ai () The proofs are obvious because 1 () a() i a i a() i a() i i1 a( 1) i1 1 1 1 1 i 1 i1 ai () [ a ( 1)] ai () 1 1 1 i1 i1 ad 1 1 ai () ai () a( 1) i1 a( 1) i1 1 a( 1) 1 1 a i a a i i1 1 1 1 i1 () [ ( 1)] () By choosig a ( ) we get the followig aalogues of (1): 7
Theorem 1 [ ] (17) 1 ad i 1 1 i [ ] (18) 1 From the recurrece relatios for the biomial coefficiets we get the well-ow formulae ad 1 1 1 1 1 1 1 1 1 1 (19) 1 ( 1)( ) 1 1 (0) For 1 it is well ow that these sums ca be used to compute 6 6 we get 1 For example from 1 1 1 1 ( 1) 1 6 6 4 4 Ufortuately i geeral the sums 1 [ ] do t have a simple expressio 1 1 For example from [ ] [] [] [] [] [ ] we get 1 1 [ ] [] [] [] [] [ ] 1 5 ( 1) 1 4 1 Now 8
( 1) ( 1) ( 1) ( 1) 1 ( 1) [ ] [ ] 1 [ ] 1 1 1 1 1 1 1 [ 1] [ ] 1 1 1 1 1 1 (1 )(1 ) [][] 1 Thus we get 1 1 1 1 1 1 5 [ ] [] [] [] [] 1 4 1 (1) This formula caot be simplified A curious geeralizatio of () is due to P Luthy [] He observed that ad more geerally 1 1 5 57911 5 19 1 5 7 9 1 5 5 () 1 1 1 () 1 1 1 1 ( ) 1 To see this observe that Similar results also hold for the aalogues of () 1 for 1 For 1 1 1 1 (4) 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 (1 ) 9
C Wheatstoe[6] also proved some other curious idetities Eg 1 0 ( 1 ) A aalogue is For 1 1 1 [][ ][ ] (5) 0 1 1 1 1 1 0 0 0 1 1 1 1 (1 )[ ] [ ] 1 (1 ) A similar idetity is m m 1 (6) 0 1 The left-had side is 1 1 1 1 1 [ ] m1 1 1 1 1 1 m (m 1) (m1) 0 0 Let us fially give a aalogue of C Wheatstoe s observatio that ad therefore 1 1 (7) 1 1 1 1 0 (8) 10
Theorem 1 1 1 1 1 1 1 [ ] (9) ad therefore 1 1 1 1 1 1 1 (0) 1 0 1 1 1 Proof Observe that (cf []) This is the special case m 0 of (6) 1 [ ] 0 (1) Therefore the left-had side of (9) becomes a ( 1) 1 a ( ) 1 a( 1) 1 a( 1) a( 1) a( ) a( ) 1 a( ) 1 1 1 1 a( 1) a( ) (1 )(1 ) a( 1) 1 a( 1) a( 1) a( ) a( ) 1 a( ) a( 1) a( ) a( 1) a( ) 1 1 1 1 1 1 1 1 if we let a ( ) This gives )(1 ) (1 )(1 ) (1 1 1 1 1 1 1 1 1 1 1 (1 )(1 ) (1 )(1 ) 1 [ ] ad therefore 1 1 1 1 1 1 1 1 0 1, which by (1) implies (0) 11
Refereces [1] K C Garrett ad K Hummel, A combiatorial proof of the sum of -cubes, Electro J Comb 11 (004), #R9 [] P Luthy, Odd sums of cosecutive odds, http://corellmathwordpresscom/008/0/15/odd-sums-of-cosecutive-odds/ [] M Schlosser, -Aalogues of the sums of cosecutive itegers, suares, cubes, uarts ad uits, Electro J Comb 11 (004), #R71 [4] B R Sears, Problem collectio, http://userstrueastlica/~brsears/math/oldprobhtm#s [5] S O Waraar, O the -aalogue of the sum of cubes, Electro J Comb 11 (004), #N1 [6] C Wheatstoe, O the formatio of powers from arithmetical progressios, Proc Royal Soc Lodo 7 (1854), 145-151 [7] G Zhao ad H Feg, A ew aalogue of the sum of cubes, Discrete Math 07 (007), 861-865 1