A203170 Sum of the fourth powers of the first n odd-indexed Fibonacci numbers.

0, 1, 17, 642, 29203, 1365539, 64107780, 3011403301, 141469813301, 6646055880582, 312223061019703, 14667837157106759, 689076118833981960, 32371909717271872585, 1520790680382055836761, 71444790066793903279242

Offset: 0

Author

Stuart Clary, Dec 30 2011

Keywords

• nonn
• easy

Comments

Natural bilateral extension (brackets mark index 0): ..., -1365539, -29203, -642, -17, -1, [0], 1, 17, 642, 29203, 1365539, ... That is, a(-n) = -a(n).

Links

• Index entries for linear recurrences with constant coefficients, signature (56,-440,770,-440,56,-1).

Crossrefs

Cf. A203169, A203171, A203172.

Cf. A001906, A103433, A163200.

Programs

Mathematica

a[n_Integer] := (1/75)(Fibonacci[8n] + 12*Fibonacci[4n] + 18 n); Table[a[n], {n, 0, 20}]

Formula

Let F(n) be the Fibonacci number A000045(n).

a(n) = Sum_{k=1..n} F(2k-1)^4.

Closed form: a(n) = (1/75)(F(8n) + 12 F(4n) + 18 n).

Recurrence: a(n) - 56 a(n-1) + 440 a(n-2) - 770 a(n-3) + 440 a(n-4) - 56 a(n-5) + a(n-6) = 0.

G.f.: A(x) = (x - 39 x^2 + 130 x^3 - 39 x^4 + x^5)/(1 - 56 x + 440 x^2 - 770 x^3 + 440 x^4 - 56 x^5 + x^6) = x(1 - 39 x + 130 x^2 - 39 x^3 + x^4)/((1 - x)^2 (1 - 7 x + x^2)(1 - 47 x + x^2)).