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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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7
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1, 9, 90, 891, 8829, 87480, 866781, 8588349, 85096170, 843160671, 8354311569, 82777250160, 820184055561, 8126651751489, 80521522263450, 797833566134451, 7905195795581109, 78327264255440040, 776092140459190341
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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.
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REFERENCES
| 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=9, q=9.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs.(39) and (45),rhs, m=9.
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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) = 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).
a(n)=Sum_{k, 0<=k<=n}8^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006
a(n)=(1/39)*[(9/2)+(3/2)*sqrt(13)]^(n+1)*sqrt(13)-(1/39)*sqrt(13)*[(9/2)-(3/2)*sqrt(13)]^(n+1), with n>=0 [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 20 2008]
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MATHEMATICA
| Join[{a=0, b=1}, Table[c=9*b+9*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
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PROG
| (Other) sage: [lucas_number1(n, 9, -9) for n in xrange(1, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]
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CROSSREFS
| Sequence in context: A052386 A186510 A158609 * A156577 A173480 A052268
Adjacent sequences: A057089 A057090 A057091 * A057093 A057094 A057095
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000
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