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A030192 Scaled Chebyshev U-polynomial evaluated at sqrt(6)/2.

%I A030192 #46 Jan 05 2025 19:51:35
%S A030192 1,6,30,144,684,3240,15336,72576,343440,1625184,7690464,36391680,
%T A030192 172207296,814893696,3856118400,18247348224,86347378944,408600184320,
%U A030192 1933516832256,9149499887616,43295898332160,204878390667264,969494954010624,4587699380060160
%N A030192 Scaled Chebyshev U-polynomial evaluated at sqrt(6)/2.
%C A030192 Binomial transform of A001834. - _Philippe Deléham_, Nov 19 2009
%H A030192 Colin Barker, <a href="/A030192/b030192.txt">Table of n, a(n) for n = 0..1000</a>
%H A030192 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=6, q=-6.
%H A030192 W. Lang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eqs. (38) and (45), lhs, m=6.
%H A030192 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A030192 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-6).
%F A030192 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
%F A030192 a(n) = (sqrt(6))^n*U(n, sqrt(6)/2).
%F A030192 G.f.: 1/(6*(x^2-x+1/6)).
%F A030192 a(2*k+1) = 6^(k+1)*A001353(k), a(2*k) = 6^k*A001834(k).
%F A030192 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
%F A030192 a(n) = Sum_{k=0..n} A109466(n,k)*6^k. - _Philippe Deléham_, Oct 28 2008
%F A030192 a(n) = ((3+sqrt(3))^n - (3-sqrt(3))^n)/sqrt(12). - Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008
%F A030192 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
%t A030192 Join[{a=1,b=6},Table[c=6*b-6*a;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 18 2011 *)
%o A030192 (Sage) [lucas_number1(n,6,6) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 22 2009
%o A030192 (PARI) a(n)=([0,1;-6,6]^n*[1;6])[1,1] \\ _Charles R Greathouse IV_, Jun 12 2015
%o A030192 (PARI) Vec(1/(6*x^2-6*x+1) + O(x^100)) \\ _Colin Barker_, Jun 15 2015
%Y A030192 Cf. A083881.
%K A030192 nonn,easy
%O A030192 0,2
%A A030192 _Wolfdieter Lang_