0,3

The only two prime a(n) are a(2) = 2 and a(4) = 37. Prime p divides a(p-1) for p = {11,19,29,31,41,59,61,71,79,89,101,109,...} = A045468[n] Primes congruent to {1, 4} mod 5. Prime p divides a((p-1)/2) for p = {29,89,101,181,229,...} = A047650[n] Primes for which golden mean tau is a quadratic residue or Primes of the form x^2 + 20y^2. 3^4 divides a(p) for p = {5,13,29,37,53,61,71,101,109,149,157,173...} = A003628[n] Primes congruent to {5, 7} mod 8. 3^5 divides a(p) for p = (37,53,109,181,197,269,397,431,541,...}. 3^6 divides a(p) for p = {109,541,...}. 3^7 divides a(p) for p = {557,...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 07 2006

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 18.

Art Benjamin and Timothy A. Carnes, Counting the Sums of Cubes of Fibonacci Numbers, to appear in Applications of Fibonacci Numbers, Volume 10, (William Webb, ed.), Kluwer Academic Publishers, 2006.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 14.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Art Benjamin and Timothy A. Carnes (paper 45), Counting the Sums of Cubes of Fibonacci Numbers.

G.f.: [1-2x-x^2]/[(1-x)(1+x-x^2)(1-4x-x^2)]. - Ralf Stephan, Apr 23 2004

a(n) = (1/2)*( F(n)*F(n+1)^2+(-1)^(n-1)*F(n-1)+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 06 2004

a(n)=sum(i=1, n, A000045(i)^3)

a(n) = (1/10)*( F(3*n+2)-(-1)^(n)*6*F(n-1)+5) (Art Benjamin and Timothy A. Carnes)

a(n+5)=4*a(n+4)+3*a(n+3)-9*a(n+2)+2*a(n+1)+a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 12 2004

(1/10) [F(3n+2) - 6(-1)^nF(n-1) + 5].

with(combinat): l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+fibonacci(i)^3; printf(`%d, `, l[i]) od:

A005968:=(-1+2*z+z**2)/(z-1)/(z**2+4*z-1)/(z**2-z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

(PARI) a(n)=(fibonacci(n)*fibonacci(n+1)^2+(-1)^(n-1)*fibonacci(n-1)+1)/2

(PARI) a(n)=(fibonacci(3*n+2)-(-1)^(n)*6*fibonacci(n-1)+5)/10

(PARI) a(n)=sum(i=1, n, fibonacci(i)^3)

Cf. A001654, A005969, A098531, A098532, A098533.

Partial sums of A056570.

Cf. A045468, A047650, A003628.

Cf. A119284, A000071, A001654, A005969, A098531, A098532, A098533, A128697.

Adjacent sequences: A005965 A005966 A005967 this_sequence A005969 A005970 A005971

Sequence in context: A034547 A124646 A124635 this_sequence A046241 A048499 A119358

nonn

njas

More terms and Maple program from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000