A057086 Scaled Chebyshev U-polynomials evaluated at sqrt(10)/2.

1, 10, 90, 800, 7100, 63000, 559000, 4960000, 44010000, 390500000, 3464900000, 30744000000, 272791000000, 2420470000000, 21476790000000, 190563200000000, 1690864100000000, 15003009000000000, 133121449000000000, 1181184400000000000, 10480629510000000000

Offset: 0

Author

Wolfdieter Lang, Aug 11 2000

Keywords

• nonn
• easy

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 sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (10,-10).

Crossrefs

Cf. A001090, A001787, A030191, A030192, A030240, A049310.

Cf. A009545, A010892, A057080, A057083, A057084, A057085.

Programs

Magma

[(10)^n*Evaluate(DicksonSecond(n, 1/10), 1): n in [0..30]]; // G. C. Greubel, May 02 2022

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 *)

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, 20)] # 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, 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..n} A109466(n,k)*10^k. - Philippe Deléham, Oct 28 2008