A030192 Scaled Chebyshev U-polynomial evaluated at sqrt(6)/2.
1, 6, 30, 144, 684, 3240, 15336, 72576, 343440, 1625184, 7690464, 36391680, 172207296, 814893696, 3856118400, 18247348224, 86347378944, 408600184320, 1933516832256, 9149499887616, 43295898332160, 204878390667264, 969494954010624, 4587699380060160
Offset: 0
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=6, q=-6.
- W. 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=6.
- Index entries for sequences related to Chebyshev polynomials.
- Index entries for linear recurrences with constant coefficients, signature (6,-6).
Crossrefs
Cf. A083881.
Programs
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Mathematica
Join[{a=1,b=6},Table[c=6*b-6*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *) -
PARI
a(n)=([0,1;-6,6]^n*[1;6])[1,1] \\ Charles R Greathouse IV, Jun 12 2015
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PARI
Vec(1/(6*x^2-6*x+1) + O(x^100)) \\ Colin Barker, Jun 15 2015
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Sage
[lucas_number1(n,6,6) for n in range(1, 21)] # Zerinvary Lajos, Apr 22 2009
Formula
a(n) = center term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 1 1 / 1 4 1 / 1 1 1]. M^n * [1 1 1] = [A083881(n) a(n) A083881(n)]. E.g., a(3) = 144 since M^3 * [1 1 1] = [54 144 54] = [A083881(3) a(3) A083881(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = (sqrt(6))^n*U(n, sqrt(6)/2).
G.f.: 1/(6*(x^2-x+1/6)).
Preceded by 0, this is the binomial transform of A001353. Its e.g.f. is then exp(3x)*sinh(sqrt(3)x)/sqrt(3). - Paul Barry, May 09 2003
a(n) = Sum_{k=0..n} A109466(n,k)*6^k. - Philippe Deléham, Oct 28 2008
a(n) = ((3+sqrt(3))^n - (3-sqrt(3))^n)/sqrt(12). - Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008
G.f.: A(x)= 1/(1-6*x+6*x^2) = G(0)/(1-3*x) where G(k) = 1 + 3*x/((1-3*x) - x*(1-3*x)/(x + (1-3*x)/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 28 2012
Comments