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A075871
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Numbers k such that 13*k^2 + 1 is a square.
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4
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0, 180, 233640, 303264540, 393637139280, 510940703520900, 663200639532988920, 860833919173116097260, 1117361763886065161254560, 1450334708690193406192321620, 1882533334518107155172472208200, 2443526817869794397220462733921980
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OFFSET
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1,2
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COMMENTS
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Limit_{n->infinity} a(n)/a(n-1) = 649 + 180*sqrt(13).
This sequence gives the values of y in solutions of the Diophantine equation x^2 - 13*y^2 = 1. The corresponding x values are in A114047. - Vincenzo Librandi, Aug 08 2010, edited by Jon E. Schoenfield, May 04 2014
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LINKS
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FORMULA
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a(n) = ((649 + 180*sqrt(13))^n - (649 - 180*sqrt(13))^n) / (2*sqrt(13)).
a(n) = 1297*(a(n-1) + a(n-2)) - a(n-3).
a(n) = 1299*(a(n-1) - a(n-2)) + a(n-3). (End)
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MATHEMATICA
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PROG
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(PARI) concat(0, Vec(180*x^2/(1-1298*x+x^2) + O(x^20))) \\ Colin Barker, Jun 13 2015
(Magma) I:=[0, 180]; [n le 2 select I[n] else 1298*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 14 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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