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A004191
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Expansion of 1/(1-12*x+x^2).
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11
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1, 12, 143, 1704, 20305, 241956, 2883167, 34356048, 409389409, 4878316860, 58130412911, 692686638072, 8254109243953, 98356624289364, 1172025382228415, 13965947962451616, 166419350167190977, 1983066254043840108, 23630375698358890319, 281581442126262843720
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Chebyshev's polynomials U(n,x) evaluated at x=6.
a(n) give all (nontrivial, integer) solutions of Pell equation b(n)^2 - 35*a(n)^2 = +1 with b(n)=A023038(n+1), n>=0.
For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 12'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, 12) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310.
a(n) = ((6+sqrt(35))^(n+1) - (6-sqrt(35))^(n+1))/(2*sqrt(35)).
a(n) = sqrt((A023038(n)^2 - 1)/35).
[A077417(n), a(n)] = the 2 X 2 matrix [1,10; 1,11]^(n+1) * [1,0]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 19 2008
a(n)=12*a(n-1)-a(n-2)for n>1, a(0)=1, a(1)=12. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2008]
a(n)=b such that (-1)^(n+1)*Integral_{x=0..Pi/2} (sin((n+1)*x))/(6+cos(x)) dx = c + b*(ln(2)+ln(3)-ln(7)). [From Francesco Daddi (francesco.daddi(AT)libero.it), Aug 01 2011]
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*11^k. - DELEHAM Philippe, Feb 10 2012
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MATHEMATICA
| lst={}; Do[AppendTo[lst, GegenbauerC[n, 1, 6]], {n, 0, 8^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 11 2008]
CoefficientList[Series[1/(1 - 12*x + x^2), {x, 0, 30}], x] (* T. D. Noe, Aug 01 2011 *)
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PROG
| sage: [lucas_number1(n, 12, 1) for n in xrange(1, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
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CROSSREFS
| Cf. A077417.
Sequence in context: A163448 A172210 A171317 * A051051 A207029 A207174
Adjacent sequences: A004188 A004189 A004190 * A004192 A004193 A004194
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KEYWORD
| nonn,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Chebyshev comments and a(n) formulas from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002
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