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A057092
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Scaled Chebyshev U-polynomials evaluated at i*3/2. Generalized Fibonacci sequence.
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12
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1, 9, 90, 891, 8829, 87480, 866781, 8588349, 85096170, 843160671, 8354311569, 82777250160, 820184055561, 8126651751489, 80521522263450, 797833566134451, 7905195795581109, 78327264255440040, 776092140459190341, 7689774642431673429, 76192801046017773930
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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->1^9, 1->(1^9)0, starting from 0. The number of 1's and 0's of this word is 9*a(n-1) and 9*a(n-2), resp.
a(n) gives the number of n-digit integers which have no digit repeated 3 times in a row. Example: a(2)= 90 which is all the 2-digit integers. a(3) = 891 = all 900 3-digit integers except 111, 222, 333, ..., 999. - Toby Gottfried, Apr 01 2013
a(n) is the number of n-digit integers which do not have two consecutive zeros. - Ran Pan, Jan 26 2016
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LINKS
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FORMULA
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a(n) = 9*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, i*3)*(-i*3)^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-9*x-9*x^2).
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MATHEMATICA
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LinearRecurrence[{9, 9}, {1, 9}, 50] (* G. C. Greubel, Jan 25 2018 *)
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PROG
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(Sage) [lucas_number1(n, 9, -9) for n in range(1, 20)] # Zerinvary Lajos, Apr 26 2009
(PARI) Vec(1/(1-9*x-9*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015
(Magma) I:=[1, 9]; [n le 2 select I[n] else 9*Self(n-1) + 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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