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A027978 a(n) = self-convolution of row n of array T given by A027960. 1

%I #12 Sep 08 2022 08:44:49

%S 1,11,42,145,473,1484,4529,13543,39870,115937,333781,953056,2702497,

%T 7618115,21365778,59657329,165926609,459905588,1270819025,3501855007,

%U 9625627686,26398369601,72248624077,197361589960,538199264833

%N a(n) = self-convolution of row n of array T given by A027960.

%H G. C. Greubel, <a href="/A027978/b027978.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).

%F From _Colin Barker_, Feb 25 2015: (Start)

%F a(n) = 5*a(n-1) - 5*a(n-2) - 5*a(n-3) + 5*a(n-4) - a(n-5).

%F G.f.: (1 +5*x -13*x^2 +8*x^3)/(1-3*x+x^2)^2. (End)

%F a(n) = 2*(n+1)*Lucas(2*n) + Fibonacci(2*n-4). - _G. C. Greubel_, Oct 01 2019

%p with(combinat); f:=fibonacci; seq(2*(n+1)*(f(2*n+1) + f(2*n-1)) + f(2*n-4), n=0..40); # _G. C. Greubel_, Oct 01 2019

%t Table[2*(n+1)*LucasL[2*n] + Fibonacci[2*n-4], {n, 0, 40}] (* _G. C. Greubel_, Oct 01 2019 *)

%o (PARI) vector(41, n, f=fibonacci; 2*n*(f(2*n-1) + f(2*n-3)) + f(2*n-6)) \\ _G. C. Greubel_, Oct 01 2019

%o (Magma) [2*(n+1)*Lucas(2*n) + Fibonacci(2*n-4): n in [0..40]]; // _G. C. Greubel_, Oct 01 2019

%o (Sage) [2*(n+1)*lucas_number2(2*n,1,-1) + fibonacci(2*n-4) for n in (0..40)] # _G. C. Greubel_, Oct 01 2019

%o (GAP) List([0..40], n-> 2*(n+1)*Lucas(1,-1,2*n)[2] + Fibonacci(2*n-4) ); # _G. C. Greubel_, Oct 01 2019

%Y Cf. A000032, A000045, A027960.

%K nonn

%O 0,2

%A _Clark Kimberling_

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Last modified March 29 09:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)