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A121740
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Solutions to the Pell equation x^2 - 17y^2 = 1 (y values).
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1
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0, 8, 528, 34840, 2298912, 151693352, 10009462320, 660472819768, 43581196642368, 2875698505576520, 189752520171407952, 12520790632807348312, 826182429245113580640, 54515519539544688973928
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| After initial term this sequence bisects A041025. See A099370 for corresponding x values. a(n+1)/a(n) apparently converges to (4+sqrt(17))^2.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..200
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Pell Equation
Index to sequences with linear recurrences with constant coefficients, signature (66,-1).
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FORMULA
| a(n) = ((33+8*sqrt(17))^(n-1) - (33-8*sqrt(17))^(n-1))/(2*sqrt(17)).
a(n) = 65*(a(n-1)+a(n-2))-a(n-3). a(n) = 67*(a(n-1)-a(n-2))+a(n-3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Feb 07 2007
a(n) = 66*a(n-1)-a(n-2) for n>1 ; a(1)=0, a(2)=8. G.f.: 8x^2/(1-66x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
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EXAMPLE
| A099370(1)^2 - 17*a(1)^2 = 33^2 - 17*8^2 = 1089 - 1088 = 1.
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MATHEMATICA
| LinearRecurrence[{66, -1}, {0, 8}, 30] (* Vincenzo Librandi, Dec 18 2011 *)
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PROG
| (PARI) Program uses fact that continued fraction for sqrt(17) = [4, 8, 8, ...]. print1("0, "); forstep(n=2, 40, 2, v=vector(n, i, if(i>1, 8, 4)); print1(contfracpnqn(v)[2, 1], ", "))
(MAGMA) I:=[0, 8]; [n le 2 select I[n] else 66*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 18 2011
(Maxima) makelist(expand(((33+8*sqrt(17))^n - (33-8*sqrt(17))^n) /(4*sqrt(17)/2)), n, 0, 16); // Vincenzo Librandi, Dec 18 2011
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CROSSREFS
| Cf. A099370, A041025, A040012.
Sequence in context: A015480 A159532 A003397 * A145182 A089671 A112035
Adjacent sequences: A121737 A121738 A121739 * A121741 A121742 A121743
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KEYWORD
| nonn,easy
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AUTHOR
| Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 31 2006
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EXTENSIONS
| Offset changed from 0 to 1 and g.f. adapted by Vincenzo Librandi, Dec 18 2011
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