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A123020 Expansion of (1 -5*x +5*x^2)/((1 -2*x)*(1 -4*x +x^2)). 1
1, 1, 2, 5, 14, 43, 142, 493, 1766, 6443, 23750, 88045, 327406, 1219531, 4546622, 16958765, 63272054, 236096683, 881049142, 3287968813, 12270563966, 45793762763, 170903438510, 637817894125, 2380363943686, 8883629492011 (list; graph; refs; listen; history; text; internal format)
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
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
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
Sequence in context: A307787 A221586 A258312 * A005317 A126566 A112808
KEYWORD
nonn,easy
AUTHOR
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

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)