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%I #92 Dec 14 2022 10:55:43
%S 1,12,143,1704,20305,241956,2883167,34356048,409389409,4878316860,
%T 58130412911,692686638072,8254109243953,98356624289364,
%U 1172025382228415,13965947962451616,166419350167190977,1983066254043840108,23630375698358890319,281581442126262843720
%N Expansion of 1/(1 - 12*x + x^2).
%C Chebyshev's polynomials U(n,x) evaluated at x=6.
%C 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.
%C 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
%C 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
%C a(n) = -a(-2-n) for all n in Z. - _Michael Somos_, Jun 29 2019
%H Vincenzo Librandi, <a href="/A004191/b004191.txt">Table of n, a(n) for n = 0..900</a>
%H K. Andersen, L. Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 12.
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-1).
%F a(n) = S(n, 12) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310.
%F a(n) = ((6+sqrt(35))^(n+1) - (6-sqrt(35))^(n+1))/(2*sqrt(35)).
%F a(n) = sqrt((A023038(n)^2 - 1)/35).
%F [A077417(n), a(n)] = the 2 X 2 matrix [1,10; 1,11]^(n+1) * [1,0]. - _Gary W. Adamson_, Mar 19 2008
%F a(n) = 12*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=12. - _Philippe Deléham_, Nov 17 2008
%F 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
%F a(n) = Sum_{k=0..n} A101950(n,k)*11^k. - _Philippe Deléham_, Feb 10 2012
%F From _Peter Bala_, Dec 23 2012 (Start):
%F Product_{n>=0} (1 + 1/a(n)) = 1/5*(5 + sqrt(35)).
%F Product_{n>=1} (1 - 1/a(n)) = 1/12*(5 + sqrt(35)). (End)
%F E.g.f.: exp(6*x)*(35*cosh(sqrt(35)*x) + 6*sqrt(35)*sinh(sqrt(35)*x))/35. - _Stefano Spezia_, Dec 14 2022
%e G.f. = 1 + 12*x + 143*x^2 + 1704*x^3 + 20305*x^4 + 241956*x^5 + ...
%p seq( simplify(ChebyshevU(n, 6)), n=0..20); # _G. C. Greubel_, Dec 23 2019
%t Table[GegenbauerC[n, 1, 6], {n,0,20}] (* _Vladimir Joseph Stephan Orlovsky_, Sep 11 2008 *)
%t CoefficientList[Series[1/(1-12*x+x^2), {x,0,30}], x] (* _T. D. Noe_, Aug 01 2011 *)
%t LinearRecurrence[{12,-1},{1,12},30] (* _Harvey P. Dale_, Feb 17 2016 *)
%t a[n_]:= ChebyshevU[n, 6]; (* _Michael Somos_, Jun 29 2019 *)
%o (Sage) [lucas_number1(n,12,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%o (Sage) [chebyshev_U(n, 6) for n in (0..20)] # _G. C. Greubel_, Dec 23 2019
%o (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
%o (PARI) Vec(1/(1-12*x+x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (PARI) {a(n) = polchebyshev(n, 2, 6)}; \\ _Michael Somos_, Jun 29 2019
%o (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
%Y Cf. A023038, A049310, A077417, A101950.
%Y 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).
%Y Cf. A323182.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Chebyshev comments and a(n) formulas from _Wolfdieter Lang_, Nov 08 2002
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