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

0, 1, 3, 15, 51, 176, 560, 1743, 5271, 15675, 45925, 133056, 381888, 1087645, 3077451, 8658951, 24245655, 67602608, 187789616, 519924075, 1435228575, 3951341811, 10852291273, 29740435200, 81340229376, 222058995001, 605201766675, 1646862596223, 4474969884411

Offset: 0

Author

Bruno Berselli, Feb 16 2017

Keywords

• nonn
• easy

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-4,-10,10,4,-5,1).

Crossrefs

Cf. A000045.

Partial sums of A169630.

Cf. A014286: partial sums of i*Fibonacci(i).

Cf. A064831: partial sums of (n+1-i)*Fibonacci(i)^2.

Programs

Magma

[&+[i*Fibonacci(i)^2: i in [0..n]]: n in [0..30]];

Maple

with(combinat): P:=proc(q) local a,n; a:=0; print(a); for n from 1 to q do
a:=a+n*fibonacci(n)^2; print(a); od; end: P(100); # Paolo P. Lava, Feb 17 2017

Mathematica

a[n_] := Sum[i*Fibonacci[i]^2, {i, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 16 2017 *)
LinearRecurrence[{5,-4,-10,10,4,-5,1},{0,1,3,15,51,176,560},30] (* Harvey P. Dale, May 15 2021 *)

Maxima

makelist(sum(i*fib(i)^2, i, 0, n), n, 0, 30);

PARI

a(n) = sum(i=0, n, i*fibonacci(i)^2) \\ Colin Barker, Feb 16 2017

Sage

[sum(i*fibonacci(i)^2 for i in [0..n]) for n in range(30)]

Formula

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).

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).

a(n) = ((n-1)*Fibonacci(n) + n*Fibonacci(n-1))*Fibonacci(n) + (1 - (-1)^n)/2.