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A078362
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A Chebyshev S-sequence with Diophantine property.
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9
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1, 13, 168, 2171, 28055, 362544, 4685017, 60542677, 782369784, 10110264515, 130651068911, 1688353631328, 21817946138353, 281944946167261, 3643466354036040, 47083117656301259, 608437063177880327
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OFFSET
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0,2
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COMMENTS
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a(n) gives the general (positive integer) solution of the Pell equation b^2 - 165*a^2 =+4 with companion sequence b(n)=A078363(n+1), n>=0.
This is the m=15 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..14 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190 and A004191. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 13's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). [John M. Campbell, Jul 08 2011]
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REFERENCES
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A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=13, q=-1.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=15.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..800
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients
Index to sequences with linear recurrences with constant coefficients, signature (13,-1).
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FORMULA
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a(n)=13*a(n-1)-a(n-2), n>= 1; a(-1)=0, a(0)=1.
a(n)=S(2*n+1, sqrt(15))/sqrt(15) = S(n, 13); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
a(n)=(ap^(n+1)-am^(n+1))/(ap-am) with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2.
G.f.: 1/(1-13*x+x^2).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*12^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/11*(11 + sqrt(165)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/26*(11 + sqrt(165)). - Peter Bala, Dec 23 2012
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MATHEMATICA
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Join[{a=1, b=13}, Table[c=13*b-a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011*)
CoefficientList[Series[1/(1 - 13 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 24 2012 *)
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PROG
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sage: [lucas_number1(n, 13, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(MAGMA) I:=[1, 13, 168]; [n le 3 select I[n] else 13*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012
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CROSSREFS
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a(n)=sqrt((A078363(n+1)^2 - 4)/165), n>=0, (Pell equation d=165, +4).
Sequence in context: A012828 A119539 A171318 * A157381 A084970 A209226
Adjacent sequences: A078359 A078360 A078361 * A078363 A078364 A078365
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Nov 29 2002
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STATUS
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approved
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