%I #60 Feb 23 2026 08:47:27

%S 0,1,9801,94109401,903638458801,8676736387298001,83314021887196947001,

%T 799981229484128697805801,7681419682192581869134354401,

%U 73756990988431941623299373152801,708214619789503821274338711878841001,6800276705461824703444258688161258139001,65296256217629821012967950649385688771846801

%N Square pentagonal numbers.

%C Limit_{n -> oo} a(n)/a(n-1) = (sqrt(2) + sqrt(3))^8 = 4801 + 1960*sqrt(6). - _Ant King_, Nov 06 2011

%C Pentagonal numbers (A000326) which are also centered octagonal numbers (A016754). - _Colin Barker_, Jan 11 2015

%D Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), pages 35-36.

%H Colin Barker, <a href="/A036353/b036353.txt">Table of n, a(n) for n = 0..252</a>

%H Muniru A. Aṣiru, <a href="https://doi.org/10.1080/0020739X.2016.1164346">All square chiliagonal numbers</a>, Int J Math Edu Sci Technol, 47:7(2016), 1123-1134.

%H Byungchan Kim, Eunmi Kim, and Jeremy Lovejoy, <a href="https://doi.org/10.1142/S1793042120400345">On weighted overpartitions related to some q-series in Ramanujan's lost notebook</a>, Int'l J. Number Theory (2021). Also at <a href="https://lovejoy.perso.math.cnrs.fr/Weightedoverpartitions_revised.pdf">Université de Paris</a> (France, 2020).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalSquareNumber.html">Pentagonal Square Number</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9603,-9603,1).

%F From _Warut Roonguthai_, Jan 05 2001: (Start)

%F a(n) = 9602*a(n-1) - a(n-2) + 200.

%F G.f.: x*(1+198*x+x^2)/((1-x)*(1-9602*x+x^2)). (End)

%F a(n+1) = 4801*a(n)+100+980*(24*a(n)^2+a(n))^(1/2). - _Richard Choulet_, Sep 21 2007

%F From _Ant King_, Nov 06 2011: (Start)

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

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

%t Table[Floor[1/96 ( Sqrt[2] + Sqrt[3] ) ^ ( 8*n - 4 ) ] , {n, 0, 9}] (* _Ant King_, Nov 06 2011 *)

%t LinearRecurrence[{9603,-9603,1},{0,1,9801,94109401},20] (* _Harvey P. Dale_, Apr 14 2019 *)

%o (PARI) for(n=0,10^9,g=(n*(3*n-1)/2); if(issquare(g),print(g)))

%o (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

%Y Intersection of A000290 and A000326.

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

%K nonn,easy

%O 0,3

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

%E More terms from _Eric W. Weisstein_