A245306 a(n) = Fibonacci(n)^2+1.

1, 2, 2, 5, 10, 26, 65, 170, 442, 1157, 3026, 7922, 20737, 54290, 142130, 372101, 974170, 2550410, 6677057, 17480762, 45765226, 119814917, 313679522, 821223650, 2149991425, 5628750626, 14736260450, 38580030725, 101003831722, 264431464442, 692290561601

Offset: 0

Author

Michel Lagneau, Jul 17 2014

Keywords

• nonn
• easy

Comments

a(n) is the product of two Fibonacci numbers.

Examples

			a(9) = Fibonacci(9)^2+1 = 34^2+1 = 1157 = A001519(4)* A001519(6)= 13*89.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Crossrefs

Cf. A000045, A001519, A001622, A007598, A080097.

Programs

Maple

with(numtheory):with(combinat,fibonacci):nn:=100:for i from 0 to nn do:x:=fibonacci(i)^2+1: printf(`%d, `,x):od:

Mathematica

Fibonacci[Range[0,30]]^2+1 (* Harvey P. Dale, Aug 05 2018 *)

PARI

a(n) = fibonacci(n)^2+1; \\ Michel Marcus, Oct 20 2020

Formula

a(2n) = A001519(n)* A001519(n+1) and a(2n+1) = A001519(n)* A001519(n+2).

a(n) = A007598(n)+1.

G.f.: -(2*x^3-4*x^2-x+1)/(x^4-3*x^3+3*x-1). - Alois P. Heinz, Jul 17 2014

Sum_{n>=0} 1/a(n) = (3 + 5*sqrt(5))/6 = (5*phi - 1)/3, where phi is the golden ratio (A001622). - Amiram Eldar, Oct 20 2020

a(2n) = A000045(2n-1)*A000045(2n+1). a(2n+1) = A000045(2n-1)*A000045(2n+3). - Chai Wah Wu, Nov 26 2020

a(n) = a(-n) = |F(n) + i|^2 for all n in Z. - Michael Somos, Sep 16 2023