A226503 Expansion of g.f. x*(1+x+x^2)/(1-x^3-x^5).

1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 7, 8, 9, 12, 13, 16, 20, 22, 28, 33, 38, 48, 55, 66, 81, 93, 114, 136, 159, 195, 229, 273, 331, 388, 468, 560, 661, 799, 948, 1129, 1359, 1609, 1928, 2307, 2738, 3287, 3916, 4666, 5594, 6654, 7953, 9510, 11320, 13547, 16164

Offset: 1

Author

Kung Fan Kai, Jun 12 2013

Keywords

• nonn
• easy

Comments

Previous name was: A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions: 1. Rabbits are able to mate at the age of two months. 2. Rabbit pairs are not fertile during their first 5 months of life, but thereafter give birth to 1 new male/female pairs at the end of every 3 month. 3. No rabbits die.

Examples

			a(6) = a(3) + a(1) = 1 + 1 = 2;
a(11) = a(8) + a(6) = 2 + 2 = 4.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1).

Programs

Mathematica

LinearRecurrence[{0, 0, 1, 0, 1}, {1, 1, 1, 1, 1}, 50] (* T. D. Noe, Jun 12 2013 *)

PARI

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,1,0,0]^(n-1)*[1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

Formula

a(n) = a(n-3) + a(n-5), a(1) = a(2) = a(3) = a(4) = a(5) = 1.

G.f.: x*(1+x+x^2)/(1-x^3-x^5). - Joerg Arndt, Jun 12 2013

Extensions

New name using g.f. from Joerg Arndt, Dec 11 2021