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A078368
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A Chebyshev S-sequence with Diophantine property.
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4
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1, 19, 360, 6821, 129239, 2448720, 46396441, 879083659, 16656193080, 315588584861, 5979526919279, 113295422881440, 2146633507828081, 40672741225852099, 770635449783361800, 14601400804658022101
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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 - 357*a^2 =+4 with companion sequence b(n)=A078369(n+1), n>=0.
This is the m=21 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..20 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364, A077412, A078366 and A049660. The m=1..3 (signed) sequences are A049347, A056594, A010892.
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 19'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=19, 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=21.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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
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a(n)=19*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n)=(ap^(n+1)-am^(n+1))/(ap-am) with ap := (19+sqrt(357))/2 and am := (19-sqrt(357))/2.
a(n)=S(2*n+1, sqrt(21))/sqrt(21) = S(n, 19); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
G.f.: 1/(1-19*x+x^2).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*18^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/17*(17 + sqrt(357)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/38*(17 + sqrt(357)). - Peter Bala, Dec 23 2012
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MATHEMATICA
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Join[{a=1, b=19}, Table[c=19*b-a; a=b; b=c, {n, 40}]] (*From Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
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PROG
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sage: [lucas_number1(n, 19, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
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a(n)=sqrt((A078369(n+1)^2 - 4)/357), n>=0, (Pell equation d=357, +4).
Cf. A077428, A078355 (Pell +4 equations).
Sequence in context: A218721 A192568 A171324 * A209227 A208504 A207877
Adjacent sequences: A078365 A078366 A078367 * A078369 A078370 A078371
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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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