OFFSET
0,2
COMMENTS
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.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
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.
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=10.
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, sqrt(10))*(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} A109466(n,k)*10^k. - Philippe Deléham, Oct 28 2008
MATHEMATICA
Join[{a=1, b=10}, Table[c=10*b-10*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 20 2011 *)
PROG
(Sage) [lucas_number1(n, 10, 10) for n in range(1, 20)] # Zerinvary Lajos, Apr 26 2009
(PARI) Vec(1/(1-10*x+10*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015
(Magma) [(10)^n*Evaluate(DicksonSecond(n, 1/10), 1): n in [0..30]]; # G. C. Greubel, May 02 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 11 2000
STATUS
approved