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A057093 Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence. 8
1, 10, 110, 1200, 13100, 143000, 1561000, 17040000, 186010000, 2030500000, 22165100000, 241956000000, 2641211000000, 28831670000000, 314728810000000, 3435604800000000, 37503336100000000, 409389409000000000, 4468927451000000000, 48783168600000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Colin Barker, Table of n, a(n) for n = 0..963

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=10, q=10.

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=10.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (10,10).

FORMULA

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).

a(n) = Sum_{k=0..n} 9^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

a(n) = -(1/70)*[5-sqrt(35)]^(n+1)*sqrt(35)+(1/70)*sqrt(35)*[5+sqrt(35)]^(n+1), with n>=0. - Paolo P. Lava, Nov 20 2008

MATHEMATICA

Join[{a=0, b=1}, Table[c=10*b+10*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)

PROG

(Sage) [lucas_number1(n, 10, -10) for n in xrange(1, 19)] # Zerinvary Lajos, Apr 26 2009

(PARI) Vec(1/(1-10*x-10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

CROSSREFS

Sequence in context: A189788 A105279 A246942 * A055276 A264915 A143749

Adjacent sequences:  A057090 A057091 A057092 * A057094 A057095 A057096

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Aug 11 2000

EXTENSIONS

Extended by T. D. Noe, May 23 2011

STATUS

approved

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Last modified May 25 05:56 EDT 2017. Contains 287012 sequences.