%I #50 Sep 21 2024 18:11:20
%S 0,1,1,64,729,15625,262144,4826809,85766121,1544804416,27680640625,
%T 496981290961,8916100448256,160005726539569,2871098559212689,
%U 51520374361000000,924491486192068809,16589354847268067929
%N Sixth power of Fibonacci numbers A000045.
%C A divisibility sequence; that is, if n divides m, then a(n) divides a(m).
%D 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).
%H Vincenzo Librandi, <a href="/A056573/b056573.txt">Table of n, a(n) for n = 0..124</a>
%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/37-40-2012/azarianIJCMS37-40-2012.pdf">Fibonacci Identities as Binomial Sums</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876. Mathematical Reviews, MR2959001. Zentralblatt MATH, Zbl 1255.05003.
%H Mohammad K. Azarian, <a href="http://www.m-hikari.com/ijcms/ijcms-2012/41-44-2012/azarianIJCMS41-44-2012.pdf">Fibonacci Identities as Binomial Sums II</a>, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059. Mathematical Reviews, MR2980853. Zentralblatt MATH, Zbl 1255.05004.
%H A. Brousseau, <a href="http://www.fq.math.ca/Scanned/6-1/brousseau3.pdf">A sequence of power formulas</a>, Fib. Quart., 6 (1968), 81-83.
%H J. Riordan, <a href="http://dx.doi.org/10.1215/S0012-7094-62-02902-2">Generating functions for powers of Fibonacci numbers</a>, Duke. Math. J. 29 (1962) 5-12.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (13,104,-260,-260,104,13,-1).
%F a(n) = F(n)^6, where F(n) = A000045(n).
%F G.f.: x*p(6, x)/q(6, x) with p(6, x) := sum_{m=0..5} A056588(5, m)*x^m = (1-x)*(1 - 11*x - 64*x^2 - 11*x^3 + x^4) and q(6, x) := sum_{m=0..7} A055870(7, m)*x^m = (1+x)*(1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2) (denominator factorization deduced from Riordan result).
%F Recursion (cf. Knuth's exercise): sum_{m=0..7} A055870(7, m)*a(n-m) = 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).
%F From _Gary Detlefs_, Jan 07 2013: (Start)
%F a(n) = (F(3*n)^2 - (-1)^n*6*F(n)*F(3*n) + 9*F(n)^2)/25.
%F a(n) = (10*F(n)^3*F(3*n) - F(3*n)^2 + 9*F(n)^2)/25. (End)
%F a(n+1) = 2*[2*F(n+1)^2-(-1)^n]^3+3*F(n)^2*F(n+1)^2*F(n+2)^2-[F(n)^6+F(n+2)^6] = {Sum(0 <= j <= [n/2]; binomial(n-j, j))}^6, for n (this is Theorem 2.2 (vi) of Azarian's second paper in the references for this sequence). - _Mohammad K. Azarian_, Jun 29 2015
%p with(combinat): A056573:=n->fibonacci(n)^6: seq(A056573(n), n=0..30); # _Wesley Ivan Hurt_, Jun 29 2015
%t f[n_]:=Fibonacci[n]^6; lst={}; Do[AppendTo[lst,f[n]],{n,0,5!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Feb 12 2010 *)
%t Fibonacci[Range[0,20]]^6 (* _Harvey P. Dale_, Sep 21 2024 *)
%o (Magma) [Fibonacci(n)^6: n in [0..20]]; // _Vincenzo Librandi_, Jun 04 2011
%o (PARI) a(n)=fibonacci(n)^6 \\ _Charles R Greathouse IV_, Jun 29 2015
%Y Cf. A000045, A007598, A055870, A056570, A056571, A056572, A056588.
%Y Sixth row of array A103323.
%K nonn,easy
%O 0,4
%A _Wolfdieter Lang_, Jul 10 2000