A056587 Tenth power of Fibonacci numbers A000045.
0, 1, 1, 1024, 59049, 9765625, 1073741824, 137858491849, 16679880978201, 2064377754059776, 253295162119140625, 31181719929966183601, 3833759992447475122176, 471584161164422542970449
Offset: 0
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..96
- 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 (89,4895,-83215,-582505,1514513,1514513,-582505,-83215,4895,89,-1).
Crossrefs
Programs
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Magma
[Fibonacci(n)^10: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011
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Mathematica
Fibonacci[Range[0,15]]^10 (* Harvey P. Dale, Jul 29 2018 *)
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PARI
a(n) = fibonacci(n)^10; \\ Michel Marcus, Sep 06 2017
Formula
a(n) = F(n)^10, F(n)=A000045(n).
G.f.: x*p(10, x)/q(10, x) with p(10, x) := sum_{m=0..9} A056588(9, m)*x^m = (1-x)*(1 - 87*x - 4047*x^2 + 42186*x^3 + 205690*x^4 + 42186*x^5 - 4047*x^6 - 87*x^7 + x^8) and q(10, x) := sum_{m=0..11} A055870(11, m)*x^m = (1+x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2)*(1 + 47*x + x^2)*(1 - 123*x + x^2) (denominator factorization deduced from Riordan result).
Recursion (cf. Knuth's exercise): sum_{m=0..11} A055870(11, m)*a(n-m) = 0, n >= 11; inputs: a(n), n=0..10. a(n) = 89*a(n-1) + 4895*a(n-2) - 83215*a(n-3) - 582505*a(n-4) + 1514513*a(n-5) + 1514513*a(n-6) - 582505*a(n-7) -83215*a(n-8) + 4895*a(n-9) + 89*a(n-10) - a(n-11).
Extensions
More terms from Larry Reeves (larryr(AT)acm.org), Jul 17 2001
Comments