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A056570 Third power of Fibonacci numbers (A000045). 31
0, 1, 1, 8, 27, 125, 512, 2197, 9261, 39304, 166375, 704969, 2985984, 12649337, 53582633, 226981000, 961504803, 4073003173, 17253512704, 73087061741, 309601747125, 1311494070536, 5555577996431, 23533806109393, 99690802348032, 422297015640625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Divisibility sequence; that is, if n divides m, then a(n) divides a(m).

REFERENCES

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).

Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop, 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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..173

Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.

Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.

A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.

Andrej Dujella, A bijective proof of Riordan's theorem on powers of Fibonacci numbers, Discrete Math. 199 (1999), no. 1-3, 217--220. MR1675924 (99k:05016).

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

J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.

H. C. Williams and R. K. Guy, Odd and even linear divisibility sequences of order 4, INTEGERS, 2015, #A33.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

FORMULA

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

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 5th row of signed Fibonomial triangle for coefficients: A055870(4, m), m=0..4

a(n) = (Fibonacci(3n) - 3(-1)^n*Fibonacci(n))/5. - Ralf 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, 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, Oct 16 2006

G.f.: x*(1-2*x-x^2)/((1+x-x^2)*(1-4*x-x^2)). - Colin Barker, Feb 28 2012

a(n) = F(n-2)*F(n+1)^2 + F(n-1)*(-1)^n. - J. M. Bergot, Mar 17 2016

a(n) = ((-3*(1/2*(-1-sqrt(5)))^n-(2-sqrt(5))^n+3*(1/2*(-1+sqrt(5)))^n+(2+sqrt(5))^n)) / (5*sqrt(5)). - Colin Barker, Jun 04 2016

EXAMPLE

a(4) = 27 because the fourth Fibonacci number is 3 and 3^3 = 27.

a(5) = 125 because the fifth Fibonacci number is 5 and 5^3 = 125.

MAPLE

A056570 := proc(n) combinat[fibonacci](n)^3 ; end proc:

seq(A056570(n), n=0..20) ;

MATHEMATICA

Table[Fibonacci[n]^3, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)

PROG

(MAGMA) [Fibonacci(n)^3: n in [0..30]]; // Vincenzo Librandi, Jun 04 2011

(PARI) a(n)=fibonacci(n)^3 \\ Charles R Greathouse IV, Sep 24 2015

(PARI) concat(0, Vec(x*(1-2*x-x^2)/((1+x-x^2)*(1-4*x-x^2)) + O(x^40))) \\ Colin Barker, Jun 04 2016

CROSSREFS

Cf. A000045, A007598, A056588, A055870, A066259, A066258.

First differences of A005968.

Third row of array A103323.

Sequence in context: A051751 A133042 A181361 * A165048 A066963 A067813

Adjacent sequences:  A056567 A056568 A056569 * A056571 A056572 A056573

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jul 10 2000

STATUS

approved

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Last modified March 29 13:17 EDT 2017. Contains 284270 sequences.