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A036353
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Square pentagonal numbers.
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7
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0, 1, 9801, 94109401, 903638458801, 8676736387298001, 83314021887196947001, 799981229484128697805801, 7681419682192581869134354401, 73756990988431941623299373152801, 708214619789503821274338711878841001, 6800276705461824703444258688161258139001
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OFFSET
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0,3
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COMMENTS
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lim(n -> Infinity, 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
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REFERENCES
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Muniru A. Asiru, All square chiliagonal numbers, International Journal of Mathematical Education in Science and Technology, Volume 47, 2016 - Issue 7; http://dx.doi.org/10.1080/0020739X.2016.1164346
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LINKS
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Colin Barker, Table of n, a(n) for n = 0..252
Eric Weisstein's World of Mathematics, Pentagonal Square Number
Index entries for linear recurrences with constant coefficients, signature (9603,-9603,1).
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FORMULA
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000326, A001078, A001079, A001110, A046172, A046173, A248205.
Sequence in context: A156735 A227489 A113937 * A174769 A031597 A031777
Adjacent sequences: A036350 A036351 A036352 * A036354 A036355 A036356
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KEYWORD
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nonn,easy
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AUTHOR
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Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)
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EXTENSIONS
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More terms from Eric W. Weisstein
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STATUS
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approved
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