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Convolution of A001654 and A007598.
0

%I #40 Jan 18 2022 11:54:38

%S 0,0,1,3,12,38,122,372,1119,3301,9624,27756,79380,225384,636061,

%T 1785639,4990116,13889618,38524238,106514652,293668923,807608137,

%U 2215854384,6066935640,16579195560,45226399440,123173004985,334955873739,909611388732,2466965351678,6682629071522

%N Convolution of A001654 and A007598.

%C Note that A001654(n) = F(n)*F(n+1) and A007598(n) = F(n)^2, for F(n) = A000045(n), the n-th Fibonacci number.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-10,0,4,-1).

%F a(n) = Sum_{i=0..n} F(i)*F(i+1)*F(n-i)^2.

%F a(n) = ((n + 2)/5)*F(n)*F(n+1) - (3/25)*(F(2*n+2) + (n + 1)*(-1)^(n + 1)).

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

%F a(n) = 4*a(n-1) - 10*a(n-3) + 4*a(n-5) - a(n-6) for n > 5. - _Amiram Eldar_, Jan 17 2022

%e For n=2, a(2) = F(0)*F(1)*F(2)^2 + F(1)*F(2)*F(1)^2 + F(2)*F(3)*F(0)^2 = 1.

%t Table[Sum[Fibonacci[i]*Fibonacci[i + 1]*Fibonacci[n - i]^2, {i, 0, n}], {n, 0, 30}]

%o (PARI) a(n) = sum(i=0, n, fibonacci(i)*fibonacci(i+1)*fibonacci(n-i)^2); \\ _Michel Marcus_, Jan 17 2022

%Y Cf. A000045, A001654, A007598.

%K nonn,easy

%O 0,4

%A _Greg Dresden_, Jan 16 2022