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A056571 Fourth power of Fibonacci numbers A000045.

%I A056571 #78 Jan 05 2025 19:51:36
%S A056571 0,1,1,16,81,625,4096,28561,194481,1336336,9150625,62742241,429981696,
%T A056571 2947295521,20200652641,138458410000,949005240561,6504586067281,
%U A056571 44583076827136,305577005139121,2094455819300625,14355614096087056
%N A056571 Fourth power of Fibonacci numbers A000045.
%C A056571 Divisibility sequence; that is, if n divides m, then a(n) divides a(m).
%C A056571 a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/4,3/4)-fences and quarter-squares (1/4 X 1 pieces, always placed so that the shorter sides are horizontal). A (w,g)-fence is a tile composed of two w X 1 pieces separated by a gap of width g. a(n+1) also equals the number of tilings of an n-board using (1/8,3/8)-fences and (1/8,7/8)-fences. - _Michael A. Allen_, Jan 11 2022
%D A056571 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 31.
%D A056571 Donald 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).
%D A056571 Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop and Abiodun A. Opanuga, Digit and Iterative Digit Sum of Fibonacci numbers, their identities and powers, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 6 (2016) pp. 4623-4627.
%H A056571 Vincenzo Librandi, <a href="/A056571/b056571.txt">Table of n, a(n) for n = 0..151</a>
%H A056571 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 A056571 Alfred Brousseau, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/6-1/brousseau3.pdf">A sequence of power formulas</a>, Fib. Quart., Vol. 6, No. 1 (1968), pp. 81-83.
%H A056571 Andrej Dujella, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00341-0">A bijective proof of Riordan's theorem on powers of Fibonacci numbers</a>, Discrete Math., Vol. 199, No. 1-3 (1999), pp. 217-220. MR1675924 (99k:05016).
%H A056571 Kenneth Edwards and Michael A. Allen, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/58-5/edwards.pdf">A new combinatorial interpretation of the Fibonacci numbers cubed</a>, Fib. Q. 58:5 (2020) 128-134.
%H A056571 Hideyuki Ohtsuka, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnNov2017.pdf">Problem N-1220</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 55, No. 4 (2017), p. 368; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnNov2018.pdf">Gelin-Cesàro Identity Yields a Telescoping Product</a>, Solution to Problem H-790 by Ramya Dutta, ibid., Vol. 56, No. 4 (2018), p. 372.
%H A056571 John 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., Vol. 29, No. 1 (1962), pp. 5-12.
%H A056571 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A056571 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).
%F A056571 a(n) = F(n)^4 = A007598(n)^2, F(n) = A000045(n).
%F A056571 G.f.: x*p(4, x)/q(4, x) with p(4, x) := sum(A056588(3, m)*x^m, m=0..3) = 1 - 4*x - 4*x^2 + x^3 = (1+x)*(1 - 5*x + x^2) and q(4, x) := Sum_{m=0..5} A055870(5, m)*x^m = 1 - 5*x - 15*x^2 + 15*x^3 + 5*x^4 - x^5 = (1-x)*(1 + 3*x + x^2)*(1 - 7*x + x^2) (denominator factorization deduced from Riordan result).
%F A056571 Recursion (cf. Knuth's exercise): 1*a(n) - 5*a(n-1) - 15*a(n-2) + 15*a(n-3) + 5*a(n-4) - 1*a(n-5) = 0, n >= 5; inputs: a(n), n=0..4.
%F A056571 (1/25)*((-1)^n*(2*F(2*n-2) - 6*F(2*n+1)) + 2*F(4*n-1) + F(4*n) + 6). - _Ralf Stephan_, May 14 2004
%F A056571 a(n) = F(n-2)*F(n-1)*F(n+1)*F(n+2) + 1 = A244855(n)+1.
%F A056571 Sum_{j=0..n} binomial(n,j)*a(j) = (3^n*A005248(n) - 4*(-1)^n*A000032(n) + 6*2^n)/25. Sum_{j=0..n} (-1)^j*binomial(n,j)*a(j) = -5^((n+1)/2-2)*(A001906(n) + 4*A000045(n)) if n odd. - _R. J. Mathar_, Oct 16 2006
%F A056571 a(n) = (F(n)*F(3n) - 3*F(n)^2*(-1)^n)/5. - _Gary Detlefs_, Dec 26 2010
%F A056571 Product_{n>=3} (1 - 1/a(n)) = phi^5/12, where phi is the golden ratio (A001622)(Ohtsuka, 2017). - _Amiram Eldar_, Dec 02 2021
%p A056571 A056571 := proc(n)
%p A056571     combinat[fibonacci](n)^4 ;
%p A056571 end proc:
%p A056571 seq(A056571(n),n=0..10) ; # _R. J. Mathar_, Jan 23 2022
%t A056571 Fibonacci[Range[0, 25]]^4 (* _Wesley Ivan Hurt_, Jan 11 2022 *)
%o A056571 (Magma) [Fibonacci(n)^4: n in [0..30]]; // _Vincenzo Librandi_, Jun 04 2011
%o A056571 (PARI) a(n)=fibonacci(n)^4 \\ _Charles R Greathouse IV_, Oct 07 2016
%Y A056571 Cf. A000045, A001622, A007598, A056570, A056588, A055870.
%Y A056571 First differences of A005969.
%Y A056571 Fourth row of array A103323.
%K A056571 nonn,easy
%O A056571 0,4
%A A056571 _Wolfdieter Lang_, Jul 10 2000