A056586 Ninth power of Fibonacci numbers A000045.

0, 1, 1, 512, 19683, 1953125, 134217728, 10604499373, 794280046581, 60716992766464, 4605366583984375, 350356403707485209, 26623333280885243904, 2023966356928852115753, 153841020405122283630137

Offset

0,4

Comments

Divisibility sequence; that is, if n divides m, then a(n) divides a(m).

References

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).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..101

A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.

J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (55,1870,-19635,-85085,136136,85085,-19635,-1870,55,1).

Formula

a(n) = F(n)^9, F(n)=A000045(n).

G.f.: x*p(9, x)/q(9, x) with p(9, x) := sum_{m=0..8} A056588(8, m)*x^m = 1 - 54*x - 1413*x^2 + 9288*x^3 + 17840*x^4 - 9288*x^5 - 1413*x^6 + 54*x^7 + x^8 and q(9, x) := sum_{m=0..10} A055870(10, m)*x^m = (1 - x - x^2)*(1 + 4*x - x^2)*(1 - 11*x - x^2)*(1 + 29*x - x^2)*(1 - 76*x - x^2) (factorization deduced from Riordan result).

Recursion (cf. Knuth's exercise): sum_{m=0..10} A055870(10, m)*a(n-m) = 0, n >= 10; inputs: a(n), n=0..9. a(n) = 55*a(n-1) + 1870*a(n-2) - 19635*a(n-3) - 85085*a(n-4) + 136136*a(n-5) + 85085*a(n-6) - 19635*a(n-7) - 1870*a(n-8) + 55*a(n-9) + a(n-10).

Prog

(Magma) [Fibonacci(n)^9: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011
(PARI) a(n) = fibonacci(n)^9; \\ Michel Marcus, Sep 06 2017

Crossrefs

Cf. A000045, A007598, A056570, A056571, A056572, A056573, A056574, A056585, A056588, A055870.

Sequence in context: A343289 A050756 A179665 * A096962 A231308 A254735

Adjacent sequences: A056583 A056584 A056585 * A056587 A056588 A056589

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 10 2000

Status

approved