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A049683 a(n) = (L(6*n) - 2)/16, where L = A000032 (the Lucas numbers). 5
0, 1, 20, 361, 6480, 116281, 2086580, 37442161, 671872320, 12056259601, 216340800500, 3882078149401, 69661065888720, 1250017107847561, 22430646875367380, 402501626648765281, 7222598632802407680, 129604273763794572961, 2325654329115499905620 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is the r = 20 member of the r-family of sequences S_r(n), n >= 1, defined in A092184 where more information can be found.

LINKS

Colin Barker, Table of n, a(n) for n = 0..750

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (19,-19,1).

FORMULA

a(n) = 1/16*{-2 + [9 + 4*sqrt(5)]^n + [9 - 4*sqrt(5)]^n}. - Ralf Stephan, Apr 14 2004

a(n) = (T(n, 9) - 1)/8 with Chebyshev's polynomials of the first kind evaluated at x = 9: T(n, 9) = A023039(n). Wolfdieter Lang, Oct 18 2004

G.f.: x*(1 + x)/((1 - x)*(1 - 18*x + x^2)) = x*(1 + x)/(1 - 19*x + 19*x^2 - x^3) (from the Stephan link, see A092184).

exp( Sum_{n >= 1} 16*a(n)*x^n/n ) = 1 + 2*Sum_{n >= 1} Fibonacci(6*n)*x^n. - Peter Bala, Jun 03 2016

a(n) = 19*a(n-1)-19*a(n-2)+a(n-3) for n>2. - Colin Barker, Jun 03 2016

MATHEMATICA

LinearRecurrence[{19, -19, 1}, {0, 1, 20}, 50] (* or *) Table[(LucasL[6*n] -2)/16, {n, 0, 30}] (* G. C. Greubel, Dec 02 2017 *)

PROG

(PARI) concat(0, Vec(x*(1+x)/((1-x)*(1-18*x+x^2)) + O(x^30))) \\ Colin Barker, Jun 03 2016

(MAGMA) [(1/16)*(-2 + (9 + 4*Sqrt(5))^n + (9 - 4*Sqrt(5))^n): n in [0..30]]; // G. C. Greubel, Dec 02 2017

CROSSREFS

Cf. A000032, A004146, A023039, A049660, A049682, A049684, A092184.

Sequence in context: A053508 A060918 A115100 * A228330 A014901 A290581

Adjacent sequences:  A049680 A049681 A049682 * A049684 A049685 A049686

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified February 21 14:40 EST 2018. Contains 299414 sequences. (Running on oeis4.)