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%I #66 Sep 08 2022 08:45:01
%S 0,1,1,32,243,3125,32768,371293,4084101,45435424,503284375,5584059449,
%T 61917364224,686719856393,7615646045657,84459630100000,
%U 936668172433707,10387823949447757,115202670521319424,1277617458486664901
%N Fifth power of Fibonacci numbers A000045.
%C 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="/A056572/b056572.txt">Table of n, a(n) for n = 0..135</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.
%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.
%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 Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop, Abiodun A. Opanuga, <a href="https://www.ripublication.com/ijaer16/ijaerv11n6_150.pdf">Digit and Iterative Digit Sum of Fibonacci numbers, their identities and powers</a>, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 6 (2016) pp 4623-4627.
%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_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,40,-60,-40,8,1).
%F a(n) = F(n)^5, F(n)=A000045(n).
%F G.f.: x*p(5, x)/q(5, x) with p(5, x) := sum(A056588(4, m)*x^m, m=0..4)= 1-7*x-16*x^2+7*x^3+x^4 and q(5, x) := sum(A055870(6, m)*x^m, m=0..6)= 1-8*x-40*x^2+60*x^3+40*x^4-8*x^5-x^6 = (1-x-x^2)*(1+4*x-x^2)*(1-11*x-x^2) (factorization deduced from Riordan result).
%F Recursion (cf. Knuth's exercise): sum(A055870(6, m)*a(n-m), m=0..6) = 0, n >= 6; inputs: a(n), n=0..5. a(n) = +8*a(n-1) +40*a(n-2) -60*a(n-3) -40*a(n-4) +8*a(n-5) +a(n-6).
%F a(n) = (10*F(n) + 5*(-1)^(n+1)*F(3*n) + F(5*n))/25, n >= 0. See the general comment on A111418 regarding the Ozeki reference; here the row 10, 5, 1 of that triangle applies. - _Wolfdieter Lang_, Aug 25 2012
%F a(n) = (F(n)^2*(F(3*n)-(-1)^n*3*F(n)))/5. - _Gary Detlefs_, Jan 07 2013
%F a(n) = F(n-2)*F(n-1)*F(n)*F(n+1)*F(n+2) + F(n). - _Tony Foster III_, Apr 11 2018
%t Table[f=Fibonacci[n];f^5,{n,0,12}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2008 *)
%t Fibonacci[Range[0,20]]^5 (* _Harvey P. Dale_, Aug 07 2022 *)
%o (Magma) [Fibonacci(n)^5: n in [0..30]]; // _Vincenzo Librandi_, Jun 04 2011
%o (PARI) a(n)=fibonacci(n)^5 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A000045, A007598, A056570-1, A056588, A055870.
%Y Fifth row of array A103323.
%K nonn,easy
%O 0,4
%A _Wolfdieter Lang_, Jul 10 2000
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