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A030192
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Scaled Chebyshev U-polynomial evaluated at sqrt(6)/2.
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16
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1, 6, 30, 144, 684, 3240, 15336, 72576, 343440, 1625184, 7690464, 36391680, 172207296, 814893696, 3856118400, 18247348224, 86347378944, 408600184320, 1933516832256, 9149499887616
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Binomial transform of A001834. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 19 2009]
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REFERENCES
| 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. 38,5 (2000) 408-419; Eqs. (38) and (45), lhs, m=6.
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LINKS
| Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| a(n) = center term in M^n * [1 1 1], where M = the 3X3 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 (qntmpkt(AT)yahoo.com), Dec 18 2004
a(n)=(sqrt(6))^n*U(n, sqrt(6)/2), g.f.: 1/(6*(x^2-x+1/6)), a(2*k+1)=6^(k+1)*A001353(k), a(2*k)=6^k*A001834(k)
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 (pbarry(AT)wit.ie), May 09 2003
a(n)=Sum_{k, 0<=k<=n} A109466(n,k)*6^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]
((3+sqrt3)^n-(3-sqrt3)^n)/sqrt12 [From Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008]
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MATHEMATICA
| Join[{a=1, b=6}, Table[c=6*b-6*a; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 18 2011*)
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PROG
| (Other) sage: [lucas_number1(n, 6, 6) for n in xrange(1, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
| Cf. A083881.
Sequence in context: A026749 A003279 A082134 * A026376 A026899 A135160
Adjacent sequences: A030189 A030190 A030191 * A030193 A030194 A030195
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KEYWORD
| nonn
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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