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A004191
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Expansion of 1/(1 - 12*x + x^2).
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32
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1, 12, 143, 1704, 20305, 241956, 2883167, 34356048, 409389409, 4878316860, 58130412911, 692686638072, 8254109243953, 98356624289364, 1172025382228415, 13965947962451616, 166419350167190977, 1983066254043840108, 23630375698358890319, 281581442126262843720
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OFFSET
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0,2
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COMMENTS
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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). - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,11}. - Milan Janjic, Jan 26 2015
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LINKS
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FORMULA
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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).
a(n) = 12*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=12. - Philippe Deléham, 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*(log(2)+log(3)-log(7)). - Francesco Daddi, Aug 01 2011
Product_{n>=0} (1 + 1/a(n)) = 1/5*(5 + sqrt(35)).
Product_{n>=1} (1 - 1/a(n)) = 1/12*(5 + sqrt(35)). (End)
E.g.f.: exp(6*x)*(35*cosh(sqrt(35)*x) + 6*sqrt(35)*sinh(sqrt(35)*x))/35. - Stefano Spezia, Dec 14 2022
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EXAMPLE
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G.f. = 1 + 12*x + 143*x^2 + 1704*x^3 + 20305*x^4 + 241956*x^5 + ...
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MAPLE
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seq( simplify(ChebyshevU(n, 6)), n=0..20); # G. C. Greubel, Dec 23 2019
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MATHEMATICA
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CoefficientList[Series[1/(1-12*x+x^2), {x, 0, 30}], x] (* T. D. Noe, Aug 01 2011 *)
LinearRecurrence[{12, -1}, {1, 12}, 30] (* Harvey P. Dale, Feb 17 2016 *)
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PROG
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(Sage) [lucas_number1(n, 12, 1) for n in range(1, 20)] # Zerinvary Lajos, Jun 25 2008
(Sage) [chebyshev_U(n, 6) for n in (0..20)] # G. C. Greubel, Dec 23 2019
(Magma) I:=[1, 12]; [n le 2 select I[n] else 12*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 13 2012
(PARI) {a(n) = polchebyshev(n, 2, 6)}; \\ Michael Somos, Jun 29 2019
(GAP) m:=8;; a:=[1, 2*m];; for n in [3..20] do a[n]:=2*m*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019
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CROSSREFS
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Chebyshev sequence U(n, m): A000027 (m=1), A001353 (m=2), A001109 (m=3), A001090 (m=4), A004189 (m=5), this sequence (m=6), A007655 (m=7), A077412 (m=8), A049660 (m=9), A075843 (m=10), A077421 (m=11), A077423 (m=12), A097309 (m=13), A097311 (m=14), A097313 (m=15), A029548 (m=16), A029547 (m=17), A144128 (m=18), A078987 (m=19), A097316 (m=33).
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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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