A071253 a(n) = n^2*(n^2+1).

0, 2, 20, 90, 272, 650, 1332, 2450, 4160, 6642, 10100, 14762, 20880, 28730, 38612, 50850, 65792, 83810, 105300, 130682, 160400, 194922, 234740, 280370, 332352, 391250, 457652, 532170, 615440, 708122, 810900, 924482, 1049600, 1187010, 1337492, 1501850, 1680912

Offset

0,2

Comments

The identity (n^5+n^3)^2+(n^2*(n^2+1))^2 = n*(n^3+n)^3 can be written as A155977(n)^2+a(n)^2 = n*A034262(n)^3. - Vincenzo Librandi, Aug 08 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

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

Formula

a(n) = A002522(n)*A000290(n). - Zerinvary Lajos, Apr 20 2008

a(n) = (1/4)*sinh(2*arcsinh(n))^2. - Artur Jasinski, Feb 10 2010

G.f.: 2*x*(1+x)*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Jan 08 2012

a(n) = A002378(A000290(n)). - Rick L. Shepherd, Sep 22 2014

Sum_{n>=1} 1/a(n) = 0.5682... = Pi^2/6- (Pi*coth Pi-1)/2 = A013661 - A259171 [J. Math. Anal. Appl. 316 (2006) 328]. - R. J. Mathar, Oct 18 2019

a(n) = 2*A037270(n). - R. J. Mathar, Oct 18 2019

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 - 1/2 + Pi*cosech(Pi)/2. - Amiram Eldar, Nov 05 2020

E.g.f.: exp(x)*x*(2 + 8*x + 6*x^2 + x^3). - Stefano Spezia, Oct 08 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Apr 16 2023

Maple

with(combinat):seq(lcm(fibonacci(3, n), n^2), n=0..35); # Zerinvary Lajos, Apr 20 2008
a:=n->add(n+add(n+add(n, j=1..n-1), j=1..n), j=1..n):seq(a(n), n=0..21); # Zerinvary Lajos, Aug 27 2008

Mathematica

Table[(1/4) Round[N[Sinh[2 ArcSinh[n]]^2, 100]], {n, 0, 10}] (* Artur Jasinski, Feb 10 2010 *)
Table[n^2*(n^2+1), {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
CoefficientList[Series[2 x (1 + x) (1 + 4 x + x^2)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

Prog

(PARI) a(n)=n^2*(n^2+1) \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A034262, A069187, A155977.

Sequence in context: A098077 A279264 A063663 * A069187 A328510 A033840

Adjacent sequences: A071250 A071251 A071252 * A071254 A071255 A071256

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 12 2002

Status

approved