A220362 -id:A220362 - cp's OEIS frontend

A220363 a(n) = Fibonacci(n)^3 + (-1)^n*Fibonacci(n+2).

1, -1, 4, 3, 35, 112, 533, 2163, 9316, 39215, 166519, 704736, 2986361, 12648727, 53583620, 226979403, 961507387, 4072998992, 17253519469, 73087050795, 309601764836, 1311494041879, 5555578042799, 23533806034368, 99690802469425, 422297015444207

Offset: 0

Michel Marcus, Dec 12 2012

An integral pentagon is a pentagon with integer sides and diagonals. There are two types of such pentagons.

Type B have sides A056570(n+2), A056570(n+2), a(n+2), A056570(n+2), A056570(n+2), and opposite diagonals A220362(n+2), A066258(n+2), A066258(n+2), A066258(n+2), A220362(n+2), for n=1,2,...

R. K. Guy, Unsolved Problems in Number Theory, D20.

Indranil Ghosh, Table of n, a(n) for n = 0..1593
J. H. Jordan, B. E. Peterson, Almost regular integer Fibonacci pentagons, Rocky Mountain J. Math. Volume 23, Number 1 (1993), 243-247.
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Table[Fibonacci[n]^3 + (-1)^n * Fibonacci[n + 2], {n, 0, 30}] (* T. D. Noe, Dec 13 2012 *)
LinearRecurrence[{3,6,-3,-1},{1,-1,4,3},30] (* Harvey P. Dale, Mar 19 2022 *)

Vec((x^2-4*x+1)/((x^2-x-1)*(x^2+4*x-1)) + O(x^100)) \\ Colin Barker, Sep 23 2014

a(n) = fibonacci(n)^3 + (-1)^n*fibonacci(n+2) \\ Charles R Greathouse IV, Feb 14 2017

a(n) = 3*a(n-1)+6*a(n-2)-3*a(n-3)-a(n-4). G.f.: (x^2-4*x+1) / ((x^2-x-1)*(x^2+4*x-1)). - Colin Barker, Sep 23 2014

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

cp's OEIS Frontend

A220363 a(n) = Fibonacci(n)^3 + (-1)^n*Fibonacci(n+2).

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

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cp's OEIS Frontend

A220363 a(n) = Fibonacci(n)^3 + (-1)^n*Fibonacci(n+2).

Views

Author

Keywords

Comments

References

Links

Programs

Mathematica

PARI

PARI

Formula

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

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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