A245679 - OEIS

%I #32 Jul 18 2022 08:19:04

%S 0,0,0,6,25,69,154,300,531,875,1364,2034,2925,4081,5550,7384,9639,

%T 12375,15656,19550,24129,29469,35650,42756,50875,60099,70524,82250,

%U 95381,110025,126294,144304,164175,186031,210000,236214,264809,295925,329706,366300

%N a(n) = pg(n, 3) + pg(n, 4) + ... + pg(n, n) where pg(n, m) is the m-th n-th-order polygonal number.

%C This is also [0, 0, 0] together with the partial sums of the terms of A005900 that are greater than 1. - _J. M. Bergot_, Jun 02 2022

%H Vincenzo Librandi, <a href="/A245679/b245679.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = (-6 - n + 2*n^2 - 2*n^3 + n^4)/6 for n>1.

%F G.f.: x^3*(x-3)*(x^2-x+2) / (x-1)^5.

%e a(5) = pg(5, 3) + pg(5, 4) + pg(5, 5) = 12 + 22 + 35 = 69.

%t CoefficientList[Series[x^3 (x - 3) (x^2 - x + 2)/(x - 1)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 01 2014 *)

%o (PARI) pg(n, m) = (m^2*(n-2)-m*(n-4))/2

%o vector(50, n, sum(m=3, n-1, pg(n-1, m)))

%Y Cf. A005900, A241452.

%K nonn,easy

%O 0,4

%A _Colin Barker_, Jul 29 2014