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A056571
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Fourth power of Fibonacci numbers A000045.
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4
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0, 1, 1, 16, 81, 625, 4096, 28561, 194481, 1336336, 9150625, 62742241, 429981696, 2947295521, 20200652641, 138458410000, 949005240561, 6504586067281, 44583076827136, 305577005139121, 2094455819300625, 14355614096087056
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Divisibility sequence; that is, if n divides m, then a(n) divides a(m).
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 31.
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.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..151
Index to divisibility sequences
Index to sequences with linear recurrences with constant coefficients, signature (5,15,-15,-5,1).
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FORMULA
| a(n)= F(n)^4, F(n)=A000045(n).
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(A055870(5, m)*x^m, m=0..5)= 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).
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.
(1/25)*((-1)^n*(2*F(2*n-2)-6*F(2*n+1)) + 2*F(4*n-1) + F(4*n) + 6). - R. Stephan, May 14 2004
F(n-2)*F(n-1)*F(n+1)*F(n+2) + 1.
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 (mathar(AT)strw.leidenuniv.nl), Oct 16 2006
a(n)=(F(n)*F(3n)-3*F(n)^2*(-1)^n)/5 [From Gary Detlefs (gdetlefs(AT)aol.com) Dec 26 2010]
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MATHEMATICA
| f[n_]:=Fibonacci[n]^4; lst={}; Do[AppendTo[lst, f[n]], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 12 2010]
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PROG
| (MAGMA) [Fibonacci(n)^4: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011
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CROSSREFS
| Cf. A000045, A007598, A056570, A056588, A055870.
First differences of A005969.
Fourth row of array A103323.
Sequence in context: A113849 A046453 A030514 * A053909 A151502 A030693
Adjacent sequences: A056568 A056569 A056570 * A056572 A056573 A056574
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KEYWORD
| nonn,easy
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jul 10 2000
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