A346513 -id:A346513 - cp's OEIS frontend

A056570 Third power of Fibonacci numbers (A000045).

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

Offset: 0

Wolfdieter Lang, Jul 10 2000

Divisibility sequence; that is, if n divides m, then a(n) divides a(m).
In general, cubing the terms of a Horadam sequence with signature (c,d) will result in a fourth-order recurrence with signature (c^3+2*c*d, c^4*d+3*(c*d)^2+2*d^3, -(c*d)^3-2*c*d^4, -d^6). - Gary Detlefs, Nov 12 2021
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using (1/3,2/3)-fences and third-squares (1/3 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 (1/6,1/3)-fences and (1/6,5/6)-fences. - Michael A. Allen, Jan 11 2022

			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.

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..173
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
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).
Kenneth Edwards and Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers cubed, Fib. Q. 58:5 (2020) 128-134.
D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux
Mariana Nagy, Simon R. Cowell and Valeriu Beiu, Survey of Cubic Fibonacci Identities - When Cuboids Carry Weight, arXiv:1902.05944 [math.HO], 2019.
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.
J. Riordan, Generating functions for powers of Fibonacci numbers, Duke. Math. J. 29 (1962) 5-12.
E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
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).

Cf. A000045, A007598, A056588, A055870, A066259, A066258.
Cf. A346513 (first differences), A005968 (partial sums).
Third row of array A103323.

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

A056570 := proc(n) combinat[fibonacci](n)^3 ; end proc:
seq(A056570(n),n=0..20) ;

Table[Fibonacci[n]^3, {n, 0, 30}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2008 *)

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

concat(0, Vec(x*(1-2*x-x^2)/((1+x-x^2)*(1-4*x-x^2)) + O(x^30))) \\ Colin Barker, Jun 04 2016

[fibonacci(n)^3 for n in (0..30)] # G. C. Greubel, Feb 20 2019

a(n) = A000045(n)^3.
G.f.: x*p(3, x)/q(3, x) with p(3, x) = Sum_{m=0..2} A056588(2, m)*x^m = 1 -2*x -x^2 and q(3, x) = Sum_{m=0..4} A055870(4, m)*x^m = 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
From R. J. Mathar, Oct 16 2006: (Start)
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. (End)
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
a(n) = F(n-1)*F(n)*F(n+1) + F(n)*(-1)^(n-1). - Tony Foster III, Apr 11 2018
5*a(n) = L(2*n-1)*F(n+2) - L(2*n+1)*F(n-2) - 7*(-1)^n*F(n), where L(n) = A000032(n). - Peter Bala, Nov 12 2019
F(n+1)*F(n)*F(n-1) = 2*Sum_{j=1..n-1} P(j)*a(n-j) for n>0, where Pell number P(n) = A000129(n). - Michael A. Allen, Jan 11 2022

A350473 a(n) = Fibonacci(n+1)^3 - Fibonacci(n-1)^3.

Original entry on oeis.org

0, 1, 7, 26, 117, 485, 2072, 8749, 37107, 157114, 665665, 2819609, 11944368, 50596649, 214331663, 907922170, 3846022173, 16292007901, 69014058568, 292348234421, 1238407008795, 5245976249306, 22222312038857, 94135224351601, 398763209531232, 1689188062337425

Offset: 0

Greg Dresden, Jan 01 2022

See A346513 for Fibonacci(n+1)^3 - Fibonacci(n)^3.

Michael De Vlieger, Table of n, a(n) for n = 0..1595
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000045, A110224, A346513.

Differences[Fibonacci[Range[-1, 26]]^3, 1, 2]

from sympy import fibonacci
def A350473(n): return fibonacci(n+1)**3-fibonacci(n-1)**3 # Chai Wah Wu, Jan 05 2022

a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4).
G.f.: x*(1 + 4*x - x^2)/(1 - 3*x - 6*x^2 + 3*x^3 + x^4).
a(n) = (4/5)*Fibonacci(3*n) + (-1)^(n)*(3/5)*Fibonacci(n).
a(n) is the numerator of the continued fraction [1,...,1, 2 ,1,...,1, 2 ,1,1,...,1] with the first two runs of 1's of length n-2 and the last run of length n-1. For example, a(4)=117 which is the numerator of the continued fraction [1,1, 2 ,1,1, 2 ,1,1,1].

A379048 Irregular triangular array: row n is the linear recurrence signature of F(i)^n - F(i-1)^n, where F = A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 2, 2, -1, 3, 6, -3, -1, 4, 19, 4, -1, 8, 40, -60, -40, 8, 1, 13, 104, -260, -260, 104, 13, -1, 21, 273, -1092, -1820, 1092, 273, -21, -1, 33, 747, -3894, -16270, -3894, 747, 33, -1, 55, 1870, -19635, -85085, 136136, 85085, -19635, -1870, 55, 1, 89

Offset: 1

Clark Kimberling, Dec 16 2024

(column 1) = (1, 2, 3, 4, 8, 13, 21, 33, 55, 89, 144, 232, ...) is linearly recurrent with signature (1,1,0,1,-1,-1).
(column 2) = (1, 2, 6, 19, 40, 104, 273, 747, 1870, 4895, 12816, 33784, ...) is linearly recurrent with signature (3, -1, 0, 8, -24, 8, 0, -8, 24, -8, 0, 1, -3, 1).
Row n is also the linear recurrence signature of L(i)^n - L(i-1)^n, where L = A000032 (Lucas numbers).

Cf. A000032, A000045, A226205, A346513.

z = 25; w[i_] := Fibonacci[i];
t[p_] := Table[w[i]^p - w[i - 1]^p, {i, 1, z}];
Column[Table[FindLinearRecurrence[t[p]], {p, 1, 12}]]  (* array *)
Flatten[Table[FindLinearRecurrence[t[p]], {p, 1, 12}]]    (* sequence *)

First 10 rows:
   1
   2      -1
   6      -3       -1
  19       4       -1
  40     -60      -40        8        1
 104    -260     -260      104       13       -1
 273   -1092    -1820     1092      273      -21      -1
 747   -3894   -16270    -3894      747       33      -1
1870  -19635   -85085   136136    85085   -19635   -1870    55   1
4895  -83215  -582505  1514513  1514513  -582505  -83215  4895  89  -1

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A056570 Third power of Fibonacci numbers (A000045).

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A350473 a(n) = Fibonacci(n+1)^3 - Fibonacci(n-1)^3.

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A379048 Irregular triangular array: row n is the linear recurrence signature of F(i)^n - F(i-1)^n, where F = A000045 (Fibonacci numbers).

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