0,2

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, A057087-92.

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.

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.

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=10.

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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<=k<=n}9^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]

(Other) sage: [lucas_number1(n, 10, -10) for n in xrange(1, 19)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]

Sequence in context: A144099 A102092 A105279 this_sequence A055276 A143749 A049398

Adjacent sequences: A057090 A057091 A057092 this_sequence A057094 A057095 A057096

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000