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 A049683 a(n) = (L(6*n) - 2)/16, where L = A000032 (the Lucas numbers). 5

%I

%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) = (L(6*n) - 2)/16, where L = A000032 (the Lucas numbers).

%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 <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) = 1/16*{-2 + [9 + 4*sqrt(5)]^n + [9 - 4*sqrt(5)]^n}. - _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

%t LinearRecurrence[{19,-19,1}, {0,1,20}, 50] (* 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 (MAGMA) [(1/16)*(-2 + (9 + 4*Sqrt(5))^n + (9 - 4*Sqrt(5))^n): n in [0..30]]; // _G. C. Greubel_, Dec 02 2017

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

%K nonn,easy

%O 0,3

%A _Clark Kimberling_

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Last modified June 19 00:55 EDT 2019. Contains 324217 sequences. (Running on oeis4.)