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%I #34 Jun 28 2023 22:21:42
%S 0,1,20,361,6480,116281,2086580,37442161,671872320,12056259601,
%T 216340800500,3882078149401,69661065888720,1250017107847561,
%U 22430646875367380,402501626648765281,7222598632802407680,129604273763794572961,2325654329115499905620
%N a(n) = (Lucas(6*n) - 2)/16.
%C 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.
%H Colin Barker, <a href="/A049683/b049683.txt">Table of n, a(n) for n = 0..750</a>
%H S. Barbero, U. Cerruti, and N. Murru, <a href="http://www.seminariomatematico.polito.it/rendiconti/78-1/BarberoCerrutiMurru.pdf">On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy</a>, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (19,-19,1).
%F a(n) = (-2 + (9 + 4*sqrt(5))^n + (9 - 4*sqrt(5))^n)/16. - _Ralf Stephan_, Apr 14 2004
%F 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
%F 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).
%F 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
%F a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3) for n>2. - _Colin Barker_, Jun 03 2016
%p with(combinat); seq( (5*fibonacci(3*n)^2 -2*(1-(-1)^n))/16, n=0..30); # _G. C. Greubel_, Dec 14 2019
%t LinearRecurrence[{19,-19,1}, {0,1,30}, 20] (* or *) Table[(LucasL[6*n] -2)/16, {n,0,30}] (* _G. C. Greubel_, Dec 02 2017 *)
%o (PARI) concat(0, Vec(x*(1+x)/((1-x)*(1-18*x+x^2)) + O(x^30))) \\ _Colin Barker_, Jun 03 2016
%o (PARI) vector(31, n, (5*fibonacci(3*n-3)^2 -2*(1+(-1)^n))/16 ) \\ _G. C. Greubel_, Dec 14 2019
%o (Magma) [(Lucas(6*n) -2)/16: n in [0..30]]; // _G. C. Greubel_, Dec 02 2017
%o (Sage) [(lucas_number2(6*n,1,-1) -2)/16 for n in (0..30)] # _G. C. Greubel_, Dec 14 2019
%o (GAP) List([0..30], n-> (Lucas(1,-1,6*n)[2] - 2)/16 ); # _G. C. Greubel_, Dec 14 2019
%Y Cf. A000032, A004146, A023039, A049660, A049682, A049684, A092184.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_
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