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A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized Fibonacci sequence. 25
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; text; internal format)
OFFSET

0,2

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, Aug 18 2003, on another version

Pisano period lengths: 1, 3, 3, 6, 1, 3, 24, 12, 9, 3, 10, 6, 56, 24, 3, 24,288, 9, 18, 6,.. - R. J. Mathar, Aug 10 2012

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..1300

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=5, q=5.

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=5.

Eric Weisstein, Horadam Sequence

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (5,5)

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, 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, 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, Nov 28 2004

a(n) = Sum_{k=0..n} 4^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

G.f.: G(0)/(2-5*x), where G(k)= 1 + 1/(1 - x*(9*k-5)/(x*(9*k+4) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

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); # Zerinvary Lajos, Dec 14 2008

MATHEMATICA

Join[{a=0, b=1}, Table[c=5*b+5*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)

PROG

(Sage) [lucas_number1(n, 5, -5) for n in xrange(1, 22)] # Zerinvary Lajos, Apr 24 2009

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: A241588 A229246 A276598 * A156195 A105481 A242157

Adjacent sequences:  A057085 A057086 A057087 * A057089 A057090 A057091

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 11 2000

STATUS

approved

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Last modified March 30 20:18 EDT 2017. Contains 284302 sequences.