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A078362
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A Chebyshev S-sequence with Diophantine property.
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8
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1, 13, 168, 2171, 28055, 362544, 4685017, 60542677, 782369784, 10110264515, 130651068911, 1688353631328, 21817946138353, 281944946167261, 3643466354036040, 47083117656301259, 608437063177880327
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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). [From John M. Campbell, Jul 08 2011]
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REFERENCES
| 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
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| 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. - DELEHAM Philippe, Feb 10 2012
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MATHEMATICA
| Join[{a=1, b=13}, Table[c=13*b-a; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 21 2011*)
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PROG
| sage: [lucas_number1(n, 13, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
| a(n)=sqrt((A078363(n+1)^2 - 4)/165), n>=0, (Pell equation d=165, +4).
Sequence in context: A012828 A119539 A171318 * A157381 A084970 A207021
Adjacent sequences: A078359 A078360 A078361 * A078363 A078364 A078365
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002
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