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A097309
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Chebyshev polynomials of the second kind, U(n,x), evaluated at x=13.
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2
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0, 1, 26, 675, 17524, 454949, 11811150, 306634951, 7960697576, 206671502025, 5365498355074, 139296285729899, 3616337930622300, 93885489910449901, 2437406399741075126, 63278680903357503375, 1642808297087554012624
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,3
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COMMENTS
| b(n)^2 - 42*(2*a(n))^2 = +1 with b(n):=A097308(n) gives all nonnegative integer solutions of this D:=42*4=168 Pell equation.
For positive n, a(n) equals the permanent of the nXn tridiagonal matrix with 26'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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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)= S(n, 26) = U(n, 13), n>=-1, with Chebyshev polynomials of 2nd kind. See A049310 for the triangle of S(n, x) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n)= ((13+2*sqrt(42))^n - (13-2*sqrt(42))^n)/(4*sqrt(42)), (Binet form).
a(n)= sum(((-1)^k)*binomial(n-k, k)*26^(n-2*k), k=0..floor(n/2)).
G.f.: 1/(1-26*x+x^2).
a(n)=26*a(n-1)-a(n-2), a(-1)=0, a(0)=1. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 18 2008]
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*25^k. - DELEHAM Philippe, Feb 10 2012
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MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 13]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
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PROG
| sage: [lucas_number1(n, 26, 1) for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
| a(n) = sqrt((A097308(n)^2 - 1)/168).
Sequence in context: A170745 A158542 A171331 * A206985 A206695 A171300
Adjacent sequences: A097306 A097307 A097308 * A097310 A097311 A097312
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Aug 31 2004
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