A057093 Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.

1, 10, 110, 1200, 13100, 143000, 1561000, 17040000, 186010000, 2030500000, 22165100000, 241956000000, 2641211000000, 28831670000000, 314728810000000, 3435604800000000, 37503336100000000, 409389409000000000, 4468927451000000000, 48783168600000000000

Offset: 0

Author

Wolfdieter Lang, Aug 11 2000

Keywords

• nonn
• easy

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, and 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
• Wolfdieter 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).

Programs

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 *)
LinearRecurrence[{10,10},{1,10},30] (* Harvey P. Dale, Jan 18 2025 *)

PARI

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

Sage

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

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

Extensions

Extended by T. D. Noe, May 23 2011