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A057093
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Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.
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10
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1, 10, 110, 1200, 13100, 143000, 1561000, 17040000, 186010000, 2030500000, 22165100000, 241956000000, 2641211000000, 28831670000000, 314728810000000, 3435604800000000, 37503336100000000, 409389409000000000, 4468927451000000000, 48783168600000000000
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OFFSET
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0,2
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COMMENTS
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This is the m=10 member of the m-family of sequences a(m,n)= S(n,i*sqrt(m))*(-i*sqrt(m))^n, with S(n,x) given in Formula and g.f.: 1/(1-m*x-m*x^2). The instances m=1..9 are A000045 (Fibonacci), A002605, A030195, A057087, A057088, A057089, A057090, A057091, A057092.
With the roots rp(m) := (m+sqrt(m*(m+4)))/2 and rm(m) := (m-sqrt(m*(m+4)))/2 the Binet form of these m-sequences is a(n,m)= (rp(m)^(n+1)-rm(m)^(n+1))/(rp(m)-rm(m)).
a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^10, 1->(1^10)0, starting from 0. The number of 1's and 0's of this word is 10*a(n-1) and 10*a(n-2), resp.
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LINKS
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FORMULA
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a(n) = 10*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, i*sqrt(10))*(-i*sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1 - 10*x - 10*x^2).
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MATHEMATICA
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PROG
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(Sage) [lucas_number1(n, 10, -10) for n in range(1, 19)] # Zerinvary Lajos, Apr 26 2009
(PARI) Vec(1/(1-10*x-10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015
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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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EXTENSIONS
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STATUS
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approved
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