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URL: https://oeis.org/A036353

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A036353
Square pentagonal numbers.
10
0, 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, 7681419682192581869134354401, 73756990988431941623299373152801, 708214619789503821274338711878841001, 6800276705461824703444258688161258139001, 65296256217629821012967950649385688771846801
OFFSET
0,3
COMMENTS
Limit_{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
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), pages 35-36.
LINKS
Muniru A. Aṣiru, All square chiliagonal numbers, Int J Math Edu Sci Technol, 47:7(2016), 1123-1134.
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.
FORMULA
From Warut Roonguthai, Jan 05 2001: (Start)
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)). (End)
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
Intersection of A000290 and A000326.
Sequence in context: A227489 A350918 A113937 * A174769 A031597 A031777
KEYWORD
nonn,easy
AUTHOR
Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)
EXTENSIONS
More terms from Eric W. Weisstein
STATUS
approved