A049683 a(n) = (Lucas(6*n) - 2)/16.

0, 1, 20, 361, 6480, 116281, 2086580, 37442161, 671872320, 12056259601, 216340800500, 3882078149401, 69661065888720, 1250017107847561, 22430646875367380, 402501626648765281, 7222598632802407680, 129604273763794572961, 2325654329115499905620

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

S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.

Index entries for sequences related to Chebyshev polynomials.

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

Formula

a(n) = (-2 + (9 + 4*sqrt(5))^n + (9 - 4*sqrt(5))^n)/16. - 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

Maple

with(combinat); seq( (5*fibonacci(3*n)^2 -2*(1-(-1)^n))/16, n=0..30); # G. C. Greubel, Dec 14 2019

Mathematica

LinearRecurrence[{19, -19, 1}, {0, 1, 30}, 20] (* 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
(PARI) vector(31, n, (5*fibonacci(3*n-3)^2 -2*(1+(-1)^n))/16 ) \\ G. C. Greubel, Dec 14 2019
(Magma) [(Lucas(6*n) -2)/16: n in [0..30]]; // G. C. Greubel, Dec 02 2017
(Sage) [(lucas_number2(6*n, 1, -1) -2)/16 for n in (0..30)] # G. C. Greubel, Dec 14 2019
(GAP) List([0..30], n-> (Lucas(1, -1, 6*n)[2] - 2)/16 ); # G. C. Greubel, Dec 14 2019

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