"Counting in their heads" - 1895 oil painting

[Albert Chan]

(08-10-2020 03:06 PM)Albert Chan Wrote:  XCas> expand(S(c,a+n-c) - S(c,a-c))    → n^3/3+a*n^2+a^2*n-n^2/2-a*n+n/6

c got *eliminated*, giving same expression as S(a,n)

We never assume variables as integer. Formula work even if they are not.

sum-of-n-squares = n*(n+1)*(2*n+1)/6 = \(\binom{2(n+1)}{3}/4\)      // 1/4 of tetrahedral numbers, side 2n

Formula is meaningless for non-integer, but the difference still gives S(a,n).

XCas> expand(simplify(comb(2*(a+n),3)/4 - comb(2*a,3)/4))    → n^3/3+a*n^2+a^2*n-n^2/2-a*n+n/6

(55² + 65² + ... + 145²) / 10825
= (5.5² + 6.5² + ... + 14.5²) / 108.25
= \( \Large {\binom{31}{3} - \binom{11}{3} \over 4×108.25}\\ \)
= (4495 - 165 ) / 433
= 10