0,2

This is the m=10 member of the m-family of sequences S(n,sqrt(m))*(sqrt(m))^n; for S(n,x) see Formula. The m=4..9 instances are A001787, A030191, A030192, A030240, A057084-5 and the m=1..3 signed sequences are A010892, A009545, A057083.

The characteristic roots are rp(m) := (m+sqrt(m*(m-4)))/2 and rm(m) := (m-sqrt(m*(m-4)))/2 and a(n,m)= (rp(m)^(n+1)-rm(m)^(n+1))/(rp(m)-rm(m)) is the Binet form of these m-sequences.

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.(38) and (45),lhs, m=10.

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, sqrt(10))*(sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.

a(2*k)= A057080(k)*10^k, a(2*k+1)=A001090(k)*10^(k+1).

G.f.: 1/(1-10*x+10*x^2).

a(n)=Sum_[k, 0<=k<=n} A109466(n,k)*10^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]

a(n)=-(1/30)*[5-sqrt(15)]^(n+1)*sqrt(15)+(1/30)*sqrt(15)*[5+sqrt(15)]^(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, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]

Sequence in context: A009454 A004985 A164552 this_sequence A092420 A010579 A010576

Adjacent sequences: A057083 A057084 A057085 this_sequence A057087 A057088 A057089

nonn,easy

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