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A057089
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Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.
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13
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1, 6, 42, 288, 1980, 13608, 93528, 642816, 4418064, 30365280, 208700064, 1434392064, 9858552768, 67757668992, 465697330560, 3200729997312, 21998563967232, 151195763787264, 1039165966526976, 7142170381885440
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OFFSET
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0,2
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COMMENTS
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a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^6, 1->(1^6)0, starting from 0. The number of 1's and 0's of this word is 6*a(n-1) and 6*a(n-2), resp.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=6, q=6.
Tanya Khovanova, Recursive Sequences
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=6.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (6,6).
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FORMULA
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a(n) = 6*(a(n-1)+6*a(n-2)), a(0)=1, a(1)=6
a(n) = S(n, i*sqrt(6))*(-i*sqrt(6))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-6*x-6*x^2).
a(n) = Sum_{k, 0<=k<=n}5^k*A063967(n,k). - Philippe Deléham, Nov 03 2006
a(n) = -(1/30)*sqrt(15)*[3-sqrt(15)]^(n+1)+(1/30)*sqrt(15)*[3+sqrt(15)]^(n+1), with n>=0. [Paolo P. Lava, Nov 20 2008]
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MATHEMATICA
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Join[{a=0, b=1}, Table[c=6*b+6*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6, 6}, {1, 6}, 40] (* Harvey P. Dale, Nov 05 2011 *)
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PROG
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(Sage) [lucas_number1(n, 6, -6) for n in xrange(1, 21)] # Zerinvary Lajos, Apr 24 2009
(MAGMA) I:=[1, 6]; [n le 2 select I[n] else 6*Self(n-1)+6*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011
(PARI) x='x+O('x^30); Vec(1/(1-6*x-6*x^2)) \\ G. C. Greubel, Jan 24 2018
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CROSSREFS
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Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.
Sequence in context: A105482 A242158 A157335 * A110711 A156361 A216517
Adjacent sequences: A057086 A057087 A057088 * A057090 A057091 A057092
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Aug 11 2000
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EXTENSIONS
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First formula corrected by Harvey P. Dale, Nov 05 2011
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STATUS
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approved
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