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A057088
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Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence.
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22
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1, 5, 30, 175, 1025, 6000, 35125, 205625, 1203750, 7046875, 41253125, 241500000, 1413765625, 8276328125, 48450468750, 283633984375, 1660422265625, 9720281250000, 56903517578125, 333118994140625, 1950112558593750
(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->11111, 1->111110, starting from 0. The number of 1's and 0's of this word is 5*a(n-1) and 5*a(n-2), resp.
a(n) / a(n-1) converges to (5 + (3 * 5^(1/2))) / 2 as n approaches infinity. (5 + (3 * 5^(1/2))) / 2 can also be written as Phi^2 + (2 * Phi), Phi^3 + Phi, Phi + 5^(1/2) + 2, (3 * Phi) + 1, (3 * Phi^2) - 2, Phi^4 - 1 and (5 + (3 * (L(n) / F(n)))) / 2, where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number as n approaches infinity. [Ross La Haye (rlahaye(AT)new.rr.com), Aug 18 2003, on another version]
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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=5, q=5.
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=5.
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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.
Eric Weisstein, Horadam Sequence
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FORMULA
| a(n) = 5*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
a(n)= S(n, i*sqrt(5))*(-i*sqrt(5))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-5*x-5*x^2).
a(n)=(1/3)*sum(k=0, n, binomial(n, k)*Fibonacci(k)*3^k) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 25 2003
a(n)=((5+3sqrt(5))/2)^n(1/2+sqrt(5)/6)+(1/2-sqrt(5)/6)((5-3sqrt(5))/2)^n - Paul Barry (pbarry(AT)wit.ie), Sep 22 2004
(a(n)) appears to be given by the floretion - 0.75'i - 0.5'j + 'k - 0.75i' + 0.5j' + 0.5k' + 1.75'ii' - 1.25'jj' + 1.75'kk' - 'ij' - 0.5'ji' - 0.75'jk' - 0.75'kj' - 1.25e ("jes") - Creighton Dement (Smith(AT)xxx.yyy.com), Nov 28 2004
a(n)=Sum_{k, 0<=k<=n}4^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006
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MAPLE
| a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=5*a[n-1]+5*a[n-2]od: seq(a[n], n=1..33); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 14 2008]
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MATHEMATICA
| Join[{a=0, b=1}, Table[c=5*b+5*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 16 2011*)
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PROG
| (Other) sage: [lucas_number1(n, 5, -5) for n in xrange(1, 22)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]
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CROSSREFS
| Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015443, A015447, A030195, A053404, A057087, A083858, A085939, A090017, A091914, A099012, A180222, A180226.
Sequence in context: A094972 A084158 A111469 * A156195 A105481 A094167
Adjacent sequences: A057085 A057086 A057087 * A057089 A057090 A057091
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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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