A110224 -id:A110224 - cp's OEIS frontend

A346513 a(n) = Fibonacci(n+1)^3 - Fibonacci(n)^3.

1, 0, 7, 19, 98, 387, 1685, 7064, 30043, 127071, 538594, 2281015, 9663353, 40933296, 173398367, 734523803, 3111498370, 13180509531, 55833549037, 236514685384, 1001892323411, 4244083925895, 17978228112962, 76156996238639, 322606213292593, 1366581849044832

Offset: 0

Lamine Ngom, Jul 21 2021

The version related to sum of consecutive Fibonacci numbers cubed is given by A110224.

a(n+1) is divisible by Fibonacci(n). The related quotient sequence is provided by A061646, from its 3rd term.

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

Cf. A056570 (partial sums).

Cf. A000045, A110224, A061646.

Differences[Fibonacci[Range[0, 26]]^3] (* Amiram Eldar, Jul 22 2021 *)

a(n) = fibonacci(n+1)^3 - fibonacci(n)^3; \\ Michel Marcus, Jul 22 2021

a(n) = F(n-1)*(2*F(n+1)^2+(-1)^(n+1)), n>0.
a(n) = F(n-1)*A061646(n+1).
G.f.: (x-1)*(x^2+2*x-1)/((x^2+4*x-1)*(x^2-x-1)). - Alois P. Heinz, Jul 21 2021
For n >= 2, a(n) is the numerator of the continued fraction [1,...,1, 3 ,1,...,1, 2 ,1,...,1] with three runs of 1's each of length n-2. For example, a(5)=387 which is the numerator of the continued fraction [1,1,1, 3 ,1,1,1, 2 ,1,1,1]. - Greg Dresden, Jan 01 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].

A226958 a(n) = Fibonacci(n-2)Fibonacci(n)Fibonacci(n+2).

Original entry on oeis.org

2, 0, 10, 24, 130, 504, 2210, 9240, 39338, 166320, 705058, 2985840, 12649570, 53582256, 226981610, 961503816, 4073004770, 17253510120, 73087065922, 309601740360, 1311494081482, 5555577978720, 23533806138050, 99690802301664, 422297015715650, 1788878864564064, 7577812474943050

Offset: 1

Ron Knott, Jun 27 2013

nonn

			a(3) = F(1)*F(3)*F(5) = 1*2*5 = 10.

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

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

Products of 3 Fibonaccis: A065563, A056570, A220362, A110224.

Table[Fibonacci[n - 2] Fibonacci[n] Fibonacci[n + 2], {n, 1, 20}]
LinearRecurrence[{3,6,-3,-1},{2,0,10,24},30] (* Harvey P. Dale, Apr 10 2022 *)
Join[{2},#[[1]]#[[3]]#[[5]]&/@Partition[Fibonacci[Range[0,40]],5,1]] (* Harvey P. Dale, May 20 2025 *)

a(n)=fibonacci(n-2)*fibonacci(n)*fibonacci(n+2); \\ Joerg Arndt, Jul 07 2013

a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4).
G.f.: 2*(1-3*x-x^2)/(1-3*x-6*x^2+3*x^3+x^4).
a(n) = Lucas(n-1)*Fibonacci(n+2) = Fibonacci(n-2)*Lucas(n+1).
a(n) = (1/5)*(Fibonacci(3*n)-8*(-1)^n*Fibonacci(n)). - Ehren Metcalfe, Mar 26 2016
For n >= 3, a(n) is the numerator of the continued fraction [1,..,1, 3 ,1,..,1, 3 ,1,..,1] with three runs of 1's each of length n-3 and each separated by a single 3. For example, a(5)=130 which is the numerator of the continued fraction [1,1, 3 ,1,1, 3 ,1,1]. - Greg Dresden, Jan 01 2022

More terms from Joerg Arndt, Jul 07 2013

A226976 Fibonacci(n)^3 + Fibonacci(n+2)^3.

Original entry on oeis.org

1, 9, 28, 133, 539, 2322, 9773, 41501, 175636, 744273, 3152359, 13354306, 56568617, 239630337, 1015087436, 4299984173, 18215017507, 77160064914, 326855259829, 1384581132277, 5865179743556, 24845300179929, 105246380344463, 445830821750018, 1888569667033489

Offset: 0

Ron Knott, Jun 27 2013

nonn

			a(2) = Fibonacci(2)^3 + Fibonacci(4)^3 = 1^3 + 2^3 = 9

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

Cf. A000045 (Fibonacci), A056570 (Fibonacci^3).

Cf. A110224 (Fib(n)^3 + Fib(n+1)^3).

Table[Fibonacci[n]^3 + Fibonacci[n+2]^3,{n,0,50}]
#[[1]]+#[[3]]&/@Partition[Fibonacci[Range[0,30]]^3,3,1] (* or *) LinearRecurrence[{3,6,-3,-1},{1,9,28,133},30] (* Harvey P. Dale, Jun 30 2025 *)

a(n) = fibonacci(n)^3+fibonacci(n+2)^3; \\ Joerg Arndt, Jul 07 2013

a(n) = 3a(n-1)+6a(n-2)-3a(n-3)-a(n-4)

G.f.: (1+6x-5x^2-2x^3)/(1-3x-6x^2+3x^3+x^4)= (2x^2+7x+1)(1-x)/((x^2-x-1)(x^2+4x-1))

cp's OEIS Frontend

A346513 a(n) = Fibonacci(n+1)^3 - Fibonacci(n)^3.

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

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A226958 a(n) = Fibonacci(n-2)*Fibonacci(n)*Fibonacci(n+2).

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A226976 Fibonacci(n)^3 + Fibonacci(n+2)^3.

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A226958 a(n) = Fibonacci(n-2)Fibonacci(n)Fibonacci(n+2).