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A282464 a(n) = Sum_{i=0..n} i*Fibonacci(i)^2.

%I A282464 #29 May 08 2025 19:41:24
%S A282464 0,1,3,15,51,176,560,1743,5271,15675,45925,133056,381888,1087645,
%T A282464 3077451,8658951,24245655,67602608,187789616,519924075,1435228575,
%U A282464 3951341811,10852291273,29740435200,81340229376,222058995001,605201766675,1646862596223,4474969884411
%N A282464 a(n) = Sum_{i=0..n} i*Fibonacci(i)^2.
%H A282464 Bruno Berselli, <a href="/A282464/b282464.txt">Table of n, a(n) for n = 0..1000</a>
%H A282464 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4,-10,10,4,-5,1).
%F A282464 O.g.f.: x*(1 - 2*x + 4*x^2 - 2*x^3 + x^4)/((1 - x)*(1 + x)^2*(1 - 3*x + x^2)^2).
%F A282464 a(n) = 5*a(n-1) - 4*a(n-2) - 10*a(n-3) + 10*a(n-4) + 4*a(n-5) - 5*a(n-6) + a(n-7).
%F A282464 a(n) = ((n-1)*Fibonacci(n) + n*Fibonacci(n-1))*Fibonacci(n) + (1 - (-1)^n)/2.
%p A282464 with(combinat): P:=proc(q) local a,n; a:=0; print(a); for n from 1 to q do
%p A282464 a:=a+n*fibonacci(n)^2; print(a); od; end: P(100); # _Paolo P. Lava_, Feb 17 2017
%t A282464 a[n_] := Sum[i*Fibonacci[i]^2, {i, 0, n}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 16 2017 *)
%t A282464 LinearRecurrence[{5,-4,-10,10,4,-5,1},{0,1,3,15,51,176,560},30] (* _Harvey P. Dale_, May 15 2021 *)
%o A282464 (PARI) a(n) = sum(i=0, n, i*fibonacci(i)^2) \\ _Colin Barker_, Feb 16 2017
%o A282464 (Sage) [sum(i*fibonacci(i)^2 for i in [0..n]) for n in range(30)]
%o A282464 (Maxima) makelist(sum(i*fib(i)^2, i, 0, n), n, 0, 30);
%o A282464 (Magma) [&+[i*Fibonacci(i)^2: i in [0..n]]: n in [0..30]];
%Y A282464 Cf. A000045.
%Y A282464 Partial sums of A169630.
%Y A282464 Cf. A014286: partial sums of i*Fibonacci(i).
%Y A282464 Cf. A064831: partial sums of (n+1-i)*Fibonacci(i)^2.
%K A282464 nonn,easy
%O A282464 0,3
%A A282464 _Bruno Berselli_, Feb 16 2017