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 A056573 Sixth power of Fibonacci numbers A000045. 3
 0, 1, 1, 64, 729, 15625, 262144, 4826809, 85766121, 1544804416, 27680640625, 496981290961, 8916100448256, 160005726539569, 2871098559212689, 51520374361000000, 924491486192068809, 16589354847268067929 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Divisibility sequence; that is, if n divides m, then a(n) divides a(m). 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..124 Index to sequences with linear recurrences with constant coefficients, signature (13,104,-260,-260,104,13,-1). FORMULA 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) MATHEMATICA f[n_]:=Fibonacci[n]^6; lst={}; Do[AppendTo[lst, f[n]], {n, 0, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Feb 12 2010] PROG (MAGMA) [Fibonacci(n)^6: n in [0..20]]; // Vincenzo Librandi, Jun 04 2011 CROSSREFS 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 KEYWORD nonn,easy AUTHOR Wolfdieter Lang Jul 10 2000 STATUS approved

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Last modified May 22 15:16 EDT 2013. Contains 225552 sequences.