A082106 Main diagonal of number array A082105.

1, 6, 33, 118, 321, 726, 1441, 2598, 4353, 6886, 10401, 15126, 21313, 29238, 39201, 51526, 66561, 84678, 106273, 131766, 161601, 196246, 236193, 281958, 334081, 393126, 459681, 534358, 617793, 710646, 813601, 927366, 1052673, 1190278, 1340961, 1505526, 1684801

Offset

0,2

Comments

4*a(n) can be written as (n^2 + 2*n + 1)^2 + (n^2 - 2*n + 1)^2 + (n^2 - 2*n - 1)^2 + (n^2 + 2*n - 1)^2. - Bruno Berselli, Jun 20 2014

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n^4 + 4*n^2 + 1.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Dec 06 2012

G.f.: (1 + x + 13*x^2 + 3*x^3 + 6*x^4)/(1 - x)^5. - Bruno Berselli, Jun 20 2014

E.g.f.: (1 + 5*x + 11*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Dec 22 2022

Sum_{n>=0} 1/a(n) = 1/2 + (Pi/4)*((1/sqrt(2)+1/sqrt(6))*coth(sqrt(2-sqrt(3))*Pi) - (1/sqrt(2)-1/sqrt(6))*coth(sqrt(2+sqrt(3))*Pi)). - Amiram Eldar, Jan 08 2023

Mathematica

Table[n^4+4n^2+1, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 6, 33, 118, 321}, 40] (* Harvey P. Dale, Dec 06 2012 *)

Prog

(Magma) [(n^2+2)^2 -3: n in [0..40]]; // G. C. Greubel, Dec 22 2022
(SageMath) [(n^2+2)^2 -3 for n in range(41)] # G. C. Greubel, Dec 22 2022

Crossrefs

Cf. A082047, A082044, A082105, A082107, A082109.

Sequence in context: A204185 A057818 A063267 * A297392 A263479 A073375

Adjacent sequences: A082103 A082104 A082105 * A082107 A082108 A082109

Keyword

nonn,easy

Author

Paul Barry, Apr 03 2003

Status

approved