A157335 Expansion of 1/( (1+x)*(1-7*x+x^2) ).

1, 6, 42, 287, 1968, 13488, 92449, 633654, 4343130, 29768255, 204034656, 1398474336, 9585285697, 65698525542, 450304393098, 3086432226143, 21154721189904, 144996616103184, 993821591532385, 6811754524623510, 46688460080832186, 320007466041201791

Offset: 0

Author

Paul Barry, Feb 27 2009

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6
• Index entries for linear recurrences with constant coefficients, signature (6,6,-1).

Crossrefs

Cf. A004187, A152119.

Programs

Magma

I:=[0, 6, 42]; [n le 3 select I[n] else 6*Self(n-1) + 6*Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 01 2018

Mathematica

CoefficientList[Series[1/((1+x)(1-7x+x^2)),{x,0,20}],x] (* or *) LinearRecurrence[{6,6,-1},{1,6,42},20] (* Harvey P. Dale, Oct 25 2014 *)

PARI

Vec(1/(x^3-6*x^2-6*x+1) + O(x^30)) \\ Colin Barker, Mar 05 2016

Formula

a(n) = A152119(n+1) * A152119(n+2).

a(n) = Sum_{k=0..floor(n/2)} (-1)^(n-k)*U(k,7/2) where U(n,x) is the n-th Chebyshev polynomial of the second kind;

a(n) = Sum_{k=0..2n} F(n+1-floor(k/2))*F(n+1-(k mod 2)-floor(k/2))*A152119(k+1).

a(n) = 7*a(n-1) - a(n-2) + (-1)^n, n > 1. - Vincenzo Librandi, Mar 13 2011

9*a(n) = 8*A004187(n+1) - A004187(n) + (-1)^n. - R. J. Mathar, Mar 15 2011

a(n)= ((-1)^(n+1)/3)*Sum_{k=0..n+1} (-1)^k*Fibonacci(4*k). - Gary Detlefs, Jan 21 2013

a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3), with a(0)=1, a(1)=6, a(2)=42. - Harvey P. Dale, Oct 25 2014

a(n) = (2^(-n)*(5*(-2)^n + (20-9*sqrt(5))*(7-3*sqrt(5))^n + (7+3*sqrt(5))^n*(20+9*sqrt(5))))/45. - Colin Barker, Mar 05 2016