A036353 Square pentagonal numbers.

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

Offset: 0

Author

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

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

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 *)

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

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)

Extensions

More terms from Eric W. Weisstein