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a(n) = Sum_{k=0..n} F(k+1)^2*F(n-k+1)^2 where F = Fibonacci numbers (A000045).
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%I #30 Oct 03 2024 15:58:02

%S 1,2,9,26,84,250,747,2182,6323,18132,51624,146004,410677,1149578,

%T 3204477,8899502,24634620,67990414,187154271,513939214,1408246247,

%U 3851081256,10512259920,28647203880,77946605545,211782868754

%N a(n) = Sum_{k=0..n} F(k+1)^2*F(n-k+1)^2 where F = Fibonacci numbers (A000045).

%C Also: the self-convolution of A007598, after A007598(0) is dropped. - _R. J. Mathar_, Aug 05 2008

%C a(n) is the number of ways to tile a 2 X (n+1) board with squares and dominoes with exactly one vertical domino. - _Greg Dresden_ and _Zijie He_, Jun 14 2022

%H Emanuele Munarini, Apr 01 2008, <a href="/A136429/b136429.txt">Table of n, a(n) for n = 0..100</a>

%H Ömer Egecioglu, Elif Saygi, and Zülfükar Saygi, <a href="https://arxiv.org/abs/2101.04740">The Mostar index of Fibonacci and Lucas cubes</a>, arXiv:2101.04740 [math.CO], 2021. Mentions this sequence.

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

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

%F Recurrence: a(n+6) = 4a(n+5) - 10a(n+3) + 4a(n+1) - a(n).

%t LinearRecurrence[{4,0,-10,0,4,-1},{1,2,9,26,84,250},30] (* _Harvey P. Dale_, Oct 03 2024 *)

%o (PARI) a(n) = sum(k=0, n, fibonacci(k+1)^2*fibonacci(n-k+1)^2); \\ _Michel Marcus_, Jan 13 2021

%Y Cf. A000045, A007598.

%K easy,nonn

%O 0,2

%A _Emanuele Munarini_, Apr 01 2008