"Counting in their heads" - 1895 oil painting
08-13-2020, 03:50 PM
Post: #22
 Albert Chan Senior Member Posts: 1,841 Joined: Jul 2018
RE: "Counting in their heads" - 1895 oil painting
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

Example, get s4(x) and s5(x) in terms of t, using divided difference.
Note: we start from x=2, instead of 0, to avoid divide-by-zero issue.

Code:
x   s4(x)    | t    s4/(2xt-t)  divided-diff 2   0+1^4=1  | 1    1/3 3   1+2^4=17 | 3    17/15       (17/15-1/3)/(3-1) = 2/5 --> s4(x) = (2xt-t) * (1/3 + (t-1)*2/5) = (2x-1)*(6t^2-t)/15 x   s5(x)    | t    s5/t^2      divided-diff 2   0+1^5=1  | 1    1 3   1+2^5=33 | 3    11/3        (11/3-1)/(3-1) = 4/3 --> s5(x) = t^2 * (1 + (t-1)*4/3) = (4t^3-t^2)/3

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
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 Messages In This Thread "Counting in their heads" - 1895 oil painting - EdS2 - 08-08-2020, 05:29 PM RE: "Counting in their heads" - 1895 oil painting - grsbanks - 08-08-2020, 07:14 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-08-2020, 07:42 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-09-2020, 12:19 PM RE: "Counting in their heads" - 1895 oil painting - EdS2 - 08-09-2020, 07:33 AM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-09-2020, 02:24 PM RE: "Counting in their heads" - 1895 oil painting - Gerson W. Barbosa - 08-09-2020, 08:11 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-10-2020, 03:06 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-10-2020, 04:33 PM RE: "Counting in their heads" - 1895 oil painting - Gerson W. Barbosa - 08-10-2020, 11:50 AM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-10-2020, 12:49 PM RE: "Counting in their heads" - 1895 oil painting - Gerson W. Barbosa - 08-11-2020, 12:33 AM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-11-2020, 10:39 AM RE: "Counting in their heads" - 1895 oil painting - Gerson W. Barbosa - 08-11-2020, 12:43 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-11-2020, 05:07 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-11-2020, 03:06 PM RE: "Counting in their heads" - 1895 oil painting - Pjwum - 08-11-2020, 08:24 AM RE: "Counting in their heads" - 1895 oil painting - EdS2 - 08-11-2020, 09:37 AM RE: "Counting in their heads" - 1895 oil painting - Gerson W. Barbosa - 08-11-2020, 11:53 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-12-2020, 01:32 AM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-14-2020, 02:25 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-12-2020, 01:50 PM RE: "Counting in their heads" - 1895 oil painting - Albert Chan - 08-13-2020 03:50 PM RE: "Counting in their heads" - 1895 oil painting - Gerson W. Barbosa - 08-15-2020, 02:55 AM RE: "Counting in their heads" - 1895 oil painting - Joe Horn - 08-15-2020, 05:04 AM RE: "Counting in their heads" - 1895 oil painting - Gerson W. Barbosa - 08-15-2020, 07:20 AM RE: "Counting in their heads" - 1895 oil painting - rprosperi - 08-14-2020, 07:20 PM

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