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A056572
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Fifth power of Fibonacci numbers A000045.
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5
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0, 1, 1, 32, 243, 3125, 32768, 371293, 4084101, 45435424, 503284375, 5584059449, 61917364224, 686719856393, 7615646045657, 84459630100000, 936668172433707, 10387823949447757, 115202670521319424, 1277617458486664901
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Divisibility sequence; that is, if n divides m, then a(n) divides a(m).
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REFERENCES
| 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
| Vincenzo Librandi, Table of n, a(n) for n = 0..135
Index to divisibility sequences
Index to sequences with linear recurrences with constant coefficients, signature (8,40,-60,-40,8,1).
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FORMULA
| a(n)= F(n)^5, F(n)=A000045(n).
G.f.: x*p(5, x)/q(5, x) with p(5, x) := sum(A056588(4, m)*x^m, m=0..4)= 1-7*x-16*x^2+7*x^3+x^4 and q(5, x) := sum(A055870(6, m)*x^m, m=0..6)= 1-8*x-40*x^2+60*x^3+40*x^4-8*x^5-x^6 = (1-x-x^2)*(1+4*x-x^2)*)*(1-11*x-x^2) (factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum(A055870(6, m)*a(n-m), m=0..6) = 0, n >= 6; inputs: a(n), n=0..5. a(n) = +8*a(n-1) +40*a(n-2) -60*a(n-3) -40*a(n-4) +8*a(n-5) +a(n-6).
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MATHEMATICA
| Table[f=Fibonacci[n]; f^5, {n, 0, 12}] (Vladimir Orlovsky, Jul 22 2008)
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PROG
| (MAGMA) [Fibonacci(n)^5: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011
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CROSSREFS
| Cf. A000045, A007598, A056570-1, A056588, A055870.
Fifth row of array A103323.
Sequence in context: A113850 A046454 A050997 * A096960 A134846 A066392
Adjacent sequences: A056569 A056570 A056571 * A056573 A056574 A056575
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jul 10 2000
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