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%I #38 Sep 08 2022 08:45:01
%S 0,1,1,256,6561,390625,16777216,815730721,37822859361,1785793904896,
%T 83733937890625,3936588805702081,184884258895036416,
%U 8686550888106661441,408066367122340274881,19170731299728100000000
%N Eighth 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="/A056585/b056585.txt">Table of n, a(n) for n = 0..107</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_09">Index entries for linear recurrences with constant coefficients</a>, signature (34,714,-4641,-12376,12376,4641,-714,-34,1).
%F a(n) = F(n)^8, F(n)=A000045(n).
%F G.f.: x*p(8, x)/q(8, x) with p(8, x) := sum_{m=0..7} A056588(7, m)*x^m = (1+x)*(1 - 34*x - 458*x^2 + 2242*x^3 - 458*x^4 - 34*x^5 + x^6) and q(8, x) := sum_{m=0..9} A055870(9, 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) (denominator factorization deduced from Riordan result).
%F Recursion (cf. Knuth's exercise): sum_{m=0..9} A055870(9, m)*a(n-m) = 0, n >= 9; inputs: a(n), n=0..8. a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).
%F a(n+1) = 8*F(n)^2*F(n+1)^2*[F(n)^4+F(n+1)^4+4*F(n)^2*F(n+1)^2+3*F(n)*F(n+1)*F(n+2)]-[F(n)^8+F(n+2)^8]+2*[2*F(n+1)^2-(-1)^n]^4 = {Sum(0 <= j <= [n/2]; binomial(n-j, j))}^8, for n>=0 (This is Theorem 2.2 (vii) of Azarian's second paper in the references for this sequence). - _Mohammad K. Azarian_, Jun 29 2015
%t lst={};Do[f=Fibonacci[n];AppendTo[lst, f^8], {n, 0, 4!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 27 2008 *)
%t Fibonacci[Range[0,20]]^8 (* _Harvey P. Dale_, Jul 03 2017 *)
%o (Magma) [Fibonacci(n)^8: n in [0..20]]; // _Vincenzo Librandi_, Jun 04 2011
%o (PARI) a(n)=fibonacci(n)^8 \\ _Charles R Greathouse IV_, Jun 30 2015
%Y Cf. A000045, A007598, A056570, A056571, A056572, A056573, A056574, A056588, A055870.
%K nonn,easy
%O 0,4
%A _Wolfdieter Lang_, Jul 10 2000
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