"Counting in their heads" - 1895 oil painting

[Gerson W. Barbosa]

(08-13-2020 03:50 PM)Albert Chan Wrote:  There is also the Faulhaber polynomials, with sum-of-powers formula as function of triangular number.

Let \(\large t = \binom{x}{2},\;{s_{2m} \over (2x-1)t}\) and \(\large {s_{2m+1} \over t^2}\) are polynomial of t, degree m-1

...

Redo previous example, using horners rule for the difference.

Code:

lua> a,b = 50, 151      -- next line replaced with t's
lua> a,b = a*(a-1)/2, b*(b-1)/2
lua> d = b-a            -- = 10100
lua> d2 = b+a           -- = (b^2-a^2)/d = 12550
lua> d3 = d2*b+a^2      -- = (b^3-a^3)/d = 143629375
lua> (4*d3 - d2)*d / 3  -- = 50^5 + 51^5 + ... + 150^5
1934166665000

Indeed there is more than a way to skin a cat (if we still may say that these days).
There is also Hurwitz zeta function.

For example,

ζ(-5, 50) - ζ(-5, 151) = 1934166665000

But what calculator has that built-in?