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

0, 288, 2400, 9408, 25920, 58080, 113568, 201600, 332928, 519840, 776160, 1117248, 1560000, 2122848, 2825760, 3690240, 4739328, 5997600, 7491168, 9247680, 11296320, 13667808, 16394400, 19509888, 23049600, 27050400, 31550688, 36590400, 42211008, 48455520

Offset: 0

Author

Seiichi Manyama, Dec 23 2018

Keywords

• nonn
• easy
• changed

Examples

			(sqrt(2) - sqrt(1))^4 = (sqrt(9) - sqrt(8))^2 = sqrt(289) - sqrt(288). So a(1) = 288.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^k: A033996(n) (k=2), A322675 (k=3), this sequence (k=4).

Cf. A180324, A185096, A339483.

Programs

Mathematica

A322677[n_] := 16*n*(n + 1)*(2*n + 1)^2; Array[A322677, 50, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 288, 2400, 9408, 25920}, 50] (* Paolo Xausa, Aug 26 2025 *)

PARI

{a(n) = 16*n*(n+1)*(2*n+1)^2}

PARI

concat(0, Vec(96*x*(3 + x)*(1 + 3*x) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Dec 23 2018

Formula

sqrt(a(n)+1) + sqrt(a(n)) = (sqrt(n+1) + sqrt(n))^4.

sqrt(a(n)+1) - sqrt(a(n)) = (sqrt(n+1) - sqrt(n))^4.

a(n) = A033996(A033996(n)).

Sum_{n>=1} 1/a(n) = (5 - Pi^2/2)/16 = 0.004074862465957543161422156253870277... - Vaclav Kotesovec, Dec 23 2018

From Colin Barker, Dec 23 2018: (Start)

G.f.: 96*x*(3 + x)*(1 + 3*x)/(1 - x)^5.

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)

From Elmo R. Oliveira, Aug 20 2025: (Start)

E.g.f.: 16*x*(2 + x)*(9 + 24*x + 4*x^2)*exp(x).

a(n) = 96*A180324(n) = 32*A339483(n) = 8*A185096(n). (End)