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 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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved