0,4

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) 5-1.

D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux,

a(n)= F(n)^3, F(n)=A000045(n).

G.f.: x*p(3, x)/q(3, x) with p(3, x) := sum(A056588(2, m)*x^m, m=0..2)= 1-2*x-x^2 and q(3, x) := sum(A055870(4, m)*x^m, m=0..4)= 1-3*x-6*x^2+3*x^3+x^4 = (1+x-x^2)*(1-4*x-x^2) (factorization deduced from Riordan result).

Recursion (cf. Knuth's exercise): 1*a(n)-3*a(n-1)-6*a(n-2)+3*a(n-3)+1*a(n-4) = 0, n >= 4, a(0)=0, a(1)=a(2)=1, a(3)=2^3. See 5-th row of signed Fibonomial triangle for coefficients: A055870(4, m), m=0..4

(1/5)[Fibonacci(3n) - 3(-1)^n*Fibonacci(n)]. - R. Stephan, May 14 2004

a(n) and a(n+1) are found as rightmost and leftmost terms (respectively) in M^n * [1 0 0 0] where M = the 4 X 4 upper triangular Pascal's triangle matrix [1 3 3 1 / 1 2 1 0 / 1 1 0 0 / 1 0 0 0]. E.g. Ma(4) = 27, a(5) = 125. M^4 * [1 0 0 0] = [125 75 45 27]; where 75 = A066259(4) and 45 = A066258(3). The characteristic polynomial of M = x^4 - 3x^3 - 6x^2 + 3x + 1. a(n)/a(n-1) of the sequence and companions tend to 2+sqrt(5) = 4.2360679...an eigenvalue of M and a root of the polynomial. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 31 2004

Sum_(j=0..n) binomial(n,j) a(j)= [2^n A001906(n)+3 A000045(n)]/5. sum_(j=0..n) (-1)^j binomial(n,j) a(j)=[(-2)^n A000045(n)-3 A001906(n)]/5. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 16 2006

with (combinat):seq(mul(fibonacci(n), k=1..3), n=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 21 2007

Cf. A000045, A007598, A056588, A055870.

Cf. A066259, A066258.

First differences of A005968.

Third row of array A103323.

Adjacent sequences: A056567 A056568 A056569 this_sequence A056571 A056572 A056573

Sequence in context: A030078 A051751 A133042 this_sequence A066963 A067813 A118720

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jul 10 2000