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A077421
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Chebyshev sequence U(n,11)=S(n,22) with Diophantine property.
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6
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1, 22, 483, 10604, 232805, 5111106, 112211527, 2463542488, 54085723209, 1187422368110, 26069206375211, 572335117886532, 12565303387128493, 275864339398940314, 6056450163389558415, 132966039255171344816
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OFFSET
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0,2
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COMMENTS
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b(n)^2 - 30*(2*a(n))^2 = 1 with the companion sequence b(n)=A077422(n+1).
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 22's along the main diagonal, and i's along the subdiagonal and the superdiagonal (i is the imaginary unit). [From John M. Campbell, Jul 08 2011]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..700
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 (22,-1).
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FORMULA
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a(n)=22*a(n-1) - a(n-1), a(-1) := 0, a(0)=1.
a(n)= S(n, 22) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the 2nd kind. See A049310.
a(n)= (ap^(n+1) - am^(n+1))/(ap - am) with ap := 11+2*sqrt(30) and am := 11-2*sqrt(30).
a(n)= sum(((-1)^k)*binomial(n-k, k)*22^(n-2*k), k=0..floor(n/2)).
a(n)=sqrt((A077422(n+1)^2-1)/30)/2.
G.f.: 1/(1-22*x+x^2). [From Philippe DELEHAM, Nov 18 2008]
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*21^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/5*(5 + sqrt(30)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/11*(5 + sqrt(30)). - Peter Bala, Dec 23 2012
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MATHEMATICA
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lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 11]], {n, 0, 8^2}]; lst [From Vladimir Joseph Stephan Orlovsky, Sep 11 2008]
CoefficientList[Series[1/(1 - 22 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, 22, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(MAGMA) I:=[1, 22, 483]; [n le 3 select I[n] else 22*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Dec 24 2012
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CROSSREFS
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Sequence in context: A139228 A158535 A171327 * A207491 A207888 A208111
Adjacent sequences: A077418 A077419 A077420 * A077422 A077423 A077424
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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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