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A056573
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Sixth power of Fibonacci numbers A000045.
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3
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0, 1, 1, 64, 729, 15625, 262144, 4826809, 85766121, 1544804416, 27680640625, 496981290961, 8916100448256, 160005726539569, 2871098559212689, 51520374361000000, 924491486192068809, 16589354847268067929
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OFFSET
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0,4
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COMMENTS
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Divisibility sequence; that is, if n divides m, then a(n) divides a(m).
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REFERENCES
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A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 1, p. 85, (exercise 1.2.8. Nr. 30) and p. 492 (solution).
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..124
Index to divisibility sequences
Index to sequences with linear recurrences with constant coefficients, signature (13,104,-260,-260,104,13,-1).
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FORMULA
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a(n)= F(n)^6, F(n)=A000045(n).
G.f.: x*p(6, x)/q(6, x) with p(6, x) := sum(A056588(5, m)*x^m, m=0..5)= (1-x)*(1-11*x-64*x^2-11*x^3+x^4) and q(6, x) := sum(A055870(7, m)*x^m, m=0..7) = (1+x)*(1-3*x+x^2)*(1+7*x+x^2)*(1-18*x+x^2) (denominator factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum(A055870(7, m)*a(n-m), m=0..7) = 0, n >= 7; inputs: a(n), n=0..6. a(n) = +13*a(n-1) +104*a(n-2) -260*a(n-3) -260*a(n-4) +104*a(n-5) +13*a(n-6) -a(n-7).
Contribution from Gary Detlefs, Jan 07 2013: (Start)
a(n) = (F(3*n)^2-(-1)^n*6*F(n)*F(3*n)+9*F(n)^2)/25.
a(n) = (10*F(n)^3*F(3*n)-F(3*n)^2+9*F(n)^2)/25. (End)
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MATHEMATICA
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f[n_]:=Fibonacci[n]^6; lst={}; Do[AppendTo[lst, f[n]], {n, 0, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Feb 12 2010]
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PROG
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(MAGMA) [Fibonacci(n)^6: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011
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CROSSREFS
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Cf. A000045, A007598, A056570-2, A056588, A055870.
Sixth row of array A103323.
Sequence in context: A046455 A092758 A030516 * A108538 A195593 A221753
Adjacent sequences: A056570 A056571 A056572 * A056574 A056575 A056576
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang Jul 10 2000
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STATUS
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approved
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