"Counting in their heads"  1895 oil painting

08142020, 02:25 PM
(This post was last modified: 08142020 02:29 PM by Albert Chan.)
Post: #23




RE: "Counting in their heads"  1895 oil painting
(08122020 01:32 AM)Albert Chan Wrote: I was wrong. There seems to be a pattern to sum of powers formula after all ... To extend formula for spacings = d, simply replace S_{p} by S_{p}/d^{p}, c by c/d Or, just check the dimensions for each term. All terms must have same units. Example, for sumofsquares, all terms should have unit of c² S2 = n*c² + (n³n)/12 * d² // c/d is dimensionless, thus c, d have same unit Example, for sum of m odd squares 1² + 3² + 5² + ... + (2m1)² // d=2, c = m = n = m*m² + m*(m²1)/12 * 2² = m*(4m²1)/3 = \(\binom{2m+1}{3}\) We can confirm this from sumofnsquares formula \(\begin{align} {n(n+1)(2n+1) \over 6} &= {n(n+1)·[(n1) + (n+2)] \over 6}\\ \binom{2n+2}{3}/4 &= \binom{n+1}{3} + \binom{n+2}{3}\\ \end{align}\) Let n = 2m: LHS = sumofmoddsquares + sumofmevensquares \(\binom{n+2}{3}\) = 4×sumofmsquares = sumofmevensquares ⇒ \(\binom{2m+1}{3}\) = sumofmoddsquares 

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