A036353 Square pentagonal numbers.

0, 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, 7681419682192581869134354401, 73756990988431941623299373152801, 708214619789503821274338711878841001, 6800276705461824703444258688161258139001

Offset

0,3

Comments

Lim_{n -> oo} a(n)/a(n-1) = (sqrt(2) + sqrt(3))^8 = 4801 + 1960*sqrt(6). - Ant King, Nov 06 2011

Pentagonal numbers (A000326) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 11 2015

Links

Colin Barker, Table of n, a(n) for n = 0..252

Muniru A. Asiru, All square chiliagonal numbers, International Journal of Mathematical Education in Science and Technology, Volume 47, 2016 - Issue 7.

Byungchan Kim, Eunmi Kim, and Jeremy Lovejoy, On weighted overpartitions related to some q-series in Ramanujan's lost notebook, Int'l J. Number Theory (2021). Also at Université de Paris (France, 2020).

Eric Weisstein's World of Mathematics, Pentagonal Square Number

Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).

Formula

a(n) = 9602*a(n-1) - a(n-2) + 200; g.f.: x*(1+198*x+x^2)/((1-x)*(1-9602*x+x^2)). - Warut Roonguthai, Jan 05 2001

a(n+1) = 4801*a(n)+100+980*(24*a(n)^2+a(n))^(1/2). - Richard Choulet, Sep 21 2007

From Ant King, Nov 06 2011: (Start)

a(n) = floor(1/96*(sqrt(2) + sqrt(3))^(8*n-4)).

a(n) = 9603*a(n-1) - 9603*a(n-2) + a(n-3).

(End)

Mathematica

Table[Floor[1/96 ( Sqrt[2] + Sqrt[3] ) ^ ( 8*n - 4 ) ] , {n, 0, 9}] (* Ant King, Nov 06 2011 *)
LinearRecurrence[{9603, -9603, 1}, {0, 1, 9801, 94109401}, 20] (* Harvey P. Dale, Apr 14 2019 *)

Prog

(PARI) for(n=0, 10^9, g=(n*(3*n-1)/2); if(issquare(g), print(g)))
(PARI) concat(0, Vec(x*(1+198*x+x^2)/((1-x)*(1-9602*x+x^2)) + O(x^20))) \\ Colin Barker, Jun 24 2015

Crossrefs

Cf. A000326, A001078, A001079, A001110, A046172, A046173, A248205.

Sequence in context: A227489 A350918 A113937 * A174769 A031597 A031777

Adjacent sequences: A036350 A036351 A036352 * A036354 A036355 A036356

Keyword

nonn,easy

Author

Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)

Extensions

More terms from Eric W. Weisstein

Status

approved