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a(n) = Sum_{1<=i<j<k<=n} F(i)*F(j)*F(k), where F(m) is the m-th Fibonacci number.
5

%I #14 Mar 03 2024 18:46:56

%S 0,0,0,2,17,102,518,2442,11010,48444,209979,902132,3854708,16416204,

%T 69769244,296148174,1256077725,5324954250,22567665834,95626443110,

%U 405154147310,1716454353240,7271524823255,30804002164872,130491325800072,552779233930872,2341634254967448,9919384305913082,42019349641680905

%N a(n) = Sum_{1<=i<j<k<=n} F(i)*F(j)*F(k), where F(m) is the m-th Fibonacci number.

%H Alois P. Heinz, <a href="/A213787/b213787.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (6, -2, -29, 16, 40, -11, -14, 2, 1).

%F G.f.: (x^4+2*x^3-4*x^2-5*x-2)*x^3 / ((x+1) * (x^2-x-1) * (x^2+4*x-1) * (x^2-3*x+1) * (x^2+x-1)). - _Alois P. Heinz_, Jun 20 2012

%p a:= n-> (Matrix(9, (i, j)-> `if`(i=j-1, 1, `if`(i=9,

%p [1, 2, -14, -11, 40, 16, -29, -2, 6][j], 0)))^(n+3).

%p <<0, -1, 0, 0, 0, 0, 2, 17, 102>>)[1, 1]:

%p seq (a(n), n=0..30); # _Alois P. Heinz_, Jun 20 2012

%t LinearRecurrence[{6, -2, -29, 16, 40, -11, -14, 2, 1}, {0, 0, 0, 2, 17, 102, 518, 2442, 11010}, 30] (* _Jean-François Alcover_, Feb 13 2016 *)

%o (PARI) x='x+O('x^50); concat([0,0,0], Vec((x^4+2*x^3-4*x^2-5*x-2)*x^3 / ((x+1) * (x^2-x-1) * (x^2+4*x-1) * (x^2-3*x+1) * (x^2+x-1)))) \\ _G. C. Greubel_, Mar 05 2017

%Y Cf. A000045, A190173.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Jun 20 2012