"Counting in their heads"  1895 oil painting

08102020, 04:33 PM
Post: #11




RE: "Counting in their heads"  1895 oil painting
(08102020 03:06 PM)Albert Chan Wrote: XCas> expand(S(c,a+nc)  S(c,ac)) → n^3/3+a*n^2+a^2*nn^2/2a*n+n/6 We never assume variables as integer. Formula work even if they are not. sumofnsquares = n*(n+1)*(2*n+1)/6 = \(\binom{2(n+1)}{3}/4\) // 1/4 of tetrahedral numbers, side 2n Formula is meaningless for noninteger, 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*nn^2/2a*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 

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