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A056574
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Seventh power of Fibonacci numbers A000045.
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4
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0, 1, 1, 128, 2187, 78125, 2097152, 62748517, 1801088541, 52523350144, 1522435234375, 44231334895529, 1283918464548864, 37281334283719577, 1082404156823183753, 31427428360210000000, 912473096871571914483
(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..115
Index to divisibility sequences
Index to sequences with linear recurrences with constant coefficients, signature (21,273,-1092,-1820,1092,273,-21,-1).
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FORMULA
| a(n)= F(n)^7, F(n)=A000045(n).
G.f.: x*p(7, x)/q(7, x) with p(7, x) := sum(A056588(6, m)*x^m, m=0..6) = 1-20*x-166*x^2+318*x^3+166*x^4-20*x^5-x^6 and q(7, x) := sum(A055870(8, m)*x^m, m=0..8) = (1+x-x^2)*(1-4*x-x^2)*(1+11*x-x^2)*(1-29*x -x^2) (factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum(A055870(8, m)*a(n-m), m=0..8) = 0, n >= 8; inputs: a(n), n=0..7. a(n) = +21*a(n-1) +273*a(n-2) -1092*a(n-3) -1820*a(n-4) +1092*a(n-5) +273*a(n-6) -21*a(n-7) -a(n-8).
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MATHEMATICA
| f[n_]:=Fibonacci[n]^7; lst={}; Do[AppendTo[lst, f[n]], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 12 2010]
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PROG
| (MAGMA) [Fibonacci(n)^7: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011
(PARI) a(n)=fibonacci(n)^7 \\ Charles R Greathouse IV, Jan 30 2012
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CROSSREFS
| Cf. A000045, A007598, A056570-3, A056588, A055870.
Seventh row of array A103323.
Sequence in context: A113852 A046456 A092759 * A096961 A019564 A128802
Adjacent sequences: A056571 A056572 A056573 * A056575 A056576 A056577
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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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