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A005968
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Sum of cubes of first n Fibonacci numbers.
(Formerly M1967)
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14
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0, 1, 2, 10, 37, 162, 674, 2871, 12132, 51436, 217811, 922780, 3908764, 16558101, 70140734, 297121734, 1258626537, 5331629710, 22585142414, 95672204155, 405273951280, 1716768021816, 7272346018247, 30806152127640, 130496954475672, 552793970116297, 2341672834801754
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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REFERENCES
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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.
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FORMULA
| 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].
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MAPLE
| 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.]
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MATHEMATICA
| f[n_]:=(Fibonacci[n]*Fibonacci[n+1]^2+(-1)^(n-1)*Fibonacci[n-1]+1)/2; Table[f[n], {n, 0, 5!}] (*From Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), 22 Nov 2010*)
Accumulate[Fibonacci[Range[0, 20]]^3]
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PROG
| (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)
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CROSSREFS
| Cf. A001654, A005969, A098531, A098532, A098533.
Partial sums of A056570.
Cf. A045468, A047650, A003628.
Cf. A119284, A000071, A001654, A005969, A098531, A098532, A098533, A128697.
Sequence in context: A034547 A124646 A124635 * A177173 A046241 A048499
Adjacent sequences: A005965 A005966 A005967 * A005969 A005970 A005971
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms and Maple program from James A. Sellers (sellersj(AT)math.psu.edu), May 29 2000
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