OFFSET
0,3
COMMENTS
Denominator of reduced g.f. is essentially the characteristic polynomial of [1, 1, 0; 1, 2, 1; 0, 1, 3]. - Paul Barry, Dec 17 2009
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Index entries for linear recurrences with constant coefficients, signature (6,-9,2).
FORMULA
From Paul Barry, Dec 17 2009: (Start)
G.f.: 1/(1 -x -x^2/(1 -2*x -x^2/(1-3*x))) = (1-5*x+5*x^2)/(1-6*x+9*x^2-2*x^3).
a(n) = ((2+sqrt(3))/6)*(2-sqrt(3))^n + ((2-sqrt(3))/6)*(2+sqrt(3))^n + 2^n/3. (End)
a(n) = (1/3)*(2^n - ChebyshevT(n+1, 2) + 4*ChebyshevT(n, 2)). - G. C. Greubel, Jul 11 2021
3*a(n) = 2^n +A001075(n-1), n>=1. - R. J. Mathar, Aug 05 2021
MATHEMATICA
Table[(2^n - ChebyshevT[n + 1, 2] + 4*ChebyshevT[n, 2])/3, {n, 0, 30}] (* G. C. Greubel, Jul 11 2021 *)
PROG
(Magma) I:=[1, 1, 2]; [n le 3 select I[n] else 6*Self(n-1) - 9*Self(n-2) +2*Self(n-3): n in [1..31]]; // G. C. Greubel, Jul 11 2021
(Sage)
def a(n): return (1/3)*(2^n - chebyshev_T(n+1, 2) + 4*chebyshev_T(n, 2))
[a(n) for n in (0..30)] # G. C. Greubel, Jul 11 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 24 2006
EXTENSIONS
Edited by N. J. A. Sloane, Jun 13 2007
New name and change of offset by G. C. Greubel, Jul 11 2021
STATUS
approved