A226591 - OEIS

%I #33 Dec 11 2021 04:55:04

%S 1,1,1,1,1,2,2,2,3,3,4,5,4,6,6,7,10,9,12,14,14,20,20,23,30,29,39,44,

%T 46,62,63,76,94,95,124,137,151,195,202,246,293,309,395,433,492,612,

%U 648,792,921,1003,1253,1374,1593,1928,2084,2537,2907,3244,3973,4379,5133,6088,6702,8103,9214,10461

%N Expansion of g.f. (x*(1+x)*(1-x+x^2)*(1+x+x^2)*(1-x^2+x^4))/(1-x^2+x^4-x^5-x^6+x^7-x^9).

%C 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 2 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. Rabbits will die after 12 months from birth.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,-1,1,1,-1,0,1).

%e For 1 <= n <= 5, a(n)=1. For 6 <= n <= 12, a(n) = a(n-3) + a(n-5). For n >= 13, a(n) = a(n-5) + a(n-8) + a(n-11).

%e From _Joerg Arndt_, Jul 06 2013: (Start)

%e G.f.: (x*(1+x)*(1-x+x^2)*(1+x+x^2)*(1-x^2+x^4))/(1-x^2+x^4-x^5-x^6+x^7-x^9);

%e a(n) = +1*a(n-2) -1*a(n-4) +1*a(n-5) +1*a(n-6) -1*a(n-7) +1*a(n-9). (End)

%t Join[{1},LinearRecurrence[{0,1,0,-1,1,1,-1,0,1},{1,1,1,1,2,2,2,3,3},65]] (* _Ray Chandler_, Jul 15 2015 *)

%o (PARI) v=[1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5]; for(n=13,100, v=concat(v,v[#v-4]+v[#v-7]+v[#v-10])); v \\ _Charles R Greathouse IV_, Jul 05 2013

%Y Cf. A226592.

%K nonn,easy

%O 1,6

%A _Emily Lu_, Jun 13 2013

%E New name using g.f. from _Joerg Arndt_, Dec 11 2021