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A004190
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Expansion of 1/(1 - 11*x + x^2).
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13
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1, 11, 120, 1309, 14279, 155760, 1699081, 18534131, 202176360, 2205405829, 24057287759, 262424759520, 2862615066961, 31226340977051, 340627135680600, 3715672151509549, 40531766530924439, 442133759688659280, 4822939590044327641, 52610201730798944771, 573889279448744064840
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OFFSET
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0,2
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COMMENTS
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Chebyshev or generalized Fibonacci sequence.
This is the m=13 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..12 (nonnegative) sequences are A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913 and A004189. The m=1..3 (signed) sequences are A049347, A056594, A010892.
All positive integer solutions of Pell equation b(n)^2 - 117*a(n)^2 = +4 together with b(n+1)=A057076(n+1), n >= 0. - Wolfdieter Lang, Aug 31 2004
For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 11'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
a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,10}. - Milan Janjic, Jan 25 2015
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LINKS
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FORMULA
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Recursion: a(n) = 11*a(n-1) - a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n) = S(2*n+1, sqrt(13))/sqrt(13) = S(n, 11); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
G.f.: 1/(1 - 11*x + x^2).
a(n) = ((11+3*sqrt(13))^(n+1) - (11-3*sqrt(13))^(n+1))/(2^(n+1)*3*sqrt(13)). - Rolf Pleisch, May 22 2011
Product_{n>=0} (1 + 1/a(n)) = 1/3*(3 + sqrt(13)).
Product_{n>=1} (1 - 1/a(n)) = 3/22*(3 + sqrt(13)). (End)
a(n) = sqrt((A057076(n+1)^2 - 4)/117).
E.g.f.: exp(11*x/2)*(39*cosh(3*sqrt(13)*x/2) + 11*sqrt(13)*sinh(3*sqrt(13)*x/2))/39. - Stefano Spezia, Aug 07 2024
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EXAMPLE
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G.f. = 1 + 11*x + 120*x^2 + 1309*x^3 + 14279*x^4 + 155760*x^5 + ...
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MAPLE
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with(combinat):seq(fibonacci(2*n+2, 3)/3, n=0..20); # Zerinvary Lajos, Apr 20 2008
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MATHEMATICA
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CoefficientList[Series[1/(1-11*x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 13 2012 *)
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PROG
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(Sage) [lucas_number1(n, 11, 1) for n in range(1, 20)] # Zerinvary Lajos, Jun 25 2008
(PARI) {a(n) = polchebyshev(n, 2, 11/2)}; /* Michael Somos, Jul 14 2018 */
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CROSSREFS
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Cf. A000027, A001090, A001109, A001353, A001906, A004187, A004189, A004254, A006190, A010892, A018913, A049310, A049347, A056594, A057076, A075835, A101950.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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