|
%I #23 Dec 25 2018 11:32:20
%S 0,288,2400,9408,25920,58080,113568,201600,332928,519840,776160,
%T 1117248,1560000,2122848,2825760,3690240,4739328,5997600,7491168,
%U 9247680,11296320,13667808,16394400,19509888,23049600,27050400,31550688,36590400,42211008,48455520
%N a(n) = 16 * n * (n+1) * (2*n+1)^2.
%H Colin Barker, <a href="/A322677/b322677.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^4.
%F sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^4.
%F a(n) = A033996(A033996(n)).
%F Sum_{n>=1} 1/a(n) = (5 - Pi^2/2)/16 = 0.004074862465957543161422156253870277... - _Vaclav Kotesovec_, Dec 23 2018
%F From _Colin Barker_, Dec 23 2018: (Start)
%F G.f.: 96*x*(3 + x)*(1 + 3*x) / (1 - x)^5.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
%F (End)
%e (sqrt(2) - sqrt(1))^4 = (sqrt(9) - sqrt(8))^2 = sqrt(289) - sqrt(288). So a(1) = 288.
%o (PARI) {a(n) = 16*n*(n+1)*(2*n+1)^2}
%o (PARI) concat(0, Vec(96*x*(3 + x)*(1 + 3*x) / (1 - x)^5 + O(x^40))) \\ _Colin Barker_, Dec 23 2018
%Y sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^k: A033996(n) (k=2), A322675 (k=3), this sequence (k=4).
%K nonn,easy
%O 0,2
%A _Seiichi Manyama_, Dec 23 2018
|
