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A322677 a(n) = 16*n*(n+1)*(2*n+1)^2.

%I A322677 #33 Aug 26 2025 14:22:20
%S A322677 0,288,2400,9408,25920,58080,113568,201600,332928,519840,776160,
%T A322677 1117248,1560000,2122848,2825760,3690240,4739328,5997600,7491168,
%U A322677 9247680,11296320,13667808,16394400,19509888,23049600,27050400,31550688,36590400,42211008,48455520
%N A322677 a(n) = 16*n*(n+1)*(2*n+1)^2.
%H A322677 Colin Barker, <a href="/A322677/b322677.txt">Table of n, a(n) for n = 0..1000</a>
%H A322677 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A322677 sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^4.
%F A322677 sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^4.
%F A322677 a(n) = A033996(A033996(n)).
%F A322677 Sum_{n>=1} 1/a(n) = (5 - Pi^2/2)/16 = 0.004074862465957543161422156253870277... - _Vaclav Kotesovec_, Dec 23 2018
%F A322677 From _Colin Barker_, Dec 23 2018: (Start)
%F A322677 G.f.: 96*x*(3 + x)*(1 + 3*x)/(1 - x)^5.
%F A322677 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4. (End)
%F A322677 From _Elmo R. Oliveira_, Aug 20 2025: (Start)
%F A322677 E.g.f.: 16*x*(2 + x)*(9 + 24*x + 4*x^2)*exp(x).
%F A322677 a(n) = 96*A180324(n) = 32*A339483(n) = 8*A185096(n). (End)
%e A322677 (sqrt(2) - sqrt(1))^4 = (sqrt(9) - sqrt(8))^2 = sqrt(289) - sqrt(288). So a(1) = 288.
%t A322677 A322677[n_] := 16*n*(n + 1)*(2*n + 1)^2; Array[A322677, 50, 0] (* or *)
%t A322677 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 288, 2400, 9408, 25920}, 50] (* _Paolo Xausa_, Aug 26 2025 *)
%o A322677 (PARI) {a(n) = 16*n*(n+1)*(2*n+1)^2}
%o A322677 (PARI) concat(0, Vec(96*x*(3 + x)*(1 + 3*x) / (1 - x)^5 + O(x^40))) \\ _Colin Barker_, Dec 23 2018
%Y A322677 sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^k: A033996(n) (k=2), A322675 (k=3), this sequence (k=4).
%Y A322677 Cf. A180324, A185096, A339483.
%K A322677 nonn,easy,changed
%O A322677 0,2
%A A322677 _Seiichi Manyama_, Dec 23 2018