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A078366
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A Chebyshev S-sequence with Diophantine property.
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7
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1, 17, 288, 4879, 82655, 1400256, 23721697, 401868593, 6808044384, 115334885935, 1953885016511, 33100710394752, 560758191694273, 9499788548407889, 160935647131239840, 2726406212682669391
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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 - 285*a^2 =+4 with companion sequence b(n)=A078367(n+1), n>=0.
This is the m=19 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..18 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364 and A077412. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 17'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
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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=17, 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=19.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index to sequences with linear recurrences with constant coefficients, signature (17,-1).
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FORMULA
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a(n)=17*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n)=S(2*n+1, sqrt(19))/sqrt(19) = S(n, 17); 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 := (17+sqrt(285))/2 and am := (17-sqrt(285))/2.
G.f.: 1/(1-17*x+x^2).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*16^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/15*(15 + sqrt(285)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/34*(15 + sqrt(285)). - Peter Bala, Dec 23 2012
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MATHEMATICA
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Join[{a=1, b=17}, Table[c=17*b-a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011*)
CoefficientList[Series[1/(1 - 17 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, 17, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(MAGMA) I:=[1, 17, 288]; [n le 3 select I[n] else 17*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((A078367(n+1)^2 - 4)/285), n>=0, (Pell equation d=285, +4).
Cf. A077428, A078355 (Pell +4 equations).
Sequence in context: A196743 A196901 A171322 * A045607 A045606 A171291
Adjacent sequences: A078363 A078364 A078365 * A078367 A078368 A078369
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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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