%I #30 Sep 08 2022 08:46:05
%S 1,1,1,1,1,3,3,3,5,5,9,11,10,18,18,26,38,36,60,68,82,130,130,192,248,
%T 272,432,472,620,876,940,1412,1700,2044,3000,3320,4608,6016,6912,
%U 10064,11792,15184,20856,23864,33432,41616,50832,71056,83344,111056,145072,172976
%N a(n) = 2*( a(n-5) + a(n-8) + a(n-11) ) for n >= 12.
%C Previous name was: Population of dying rabbits: Rabbit pairs are not fertile during their first 5 months of life, but thereafter give birth to 2 new male/female pairs at the end of every 3 month. Rabbits will die after 12 months from birth.
%C The Hoggatt-Lind article shows that the birth polynomial is B(x) = 2*x^5+2*x^8+2*x^11, that the death polynomial is D(x)=x^12, and the total number of rabbit pairs, a(n), has the generating function T(x) = (1-D(x)) / ((1-x)*(1-B(x)) = (1-x^12) / ((1-x)*(1-2x^5-2x^8-2x^11)) = (x+1) *(x^2+1) *(x^2+x+1) *(x^2-x+1) *(x^4-x^2+1) / (1-2x^5-2x^8-2x^11). - _R. J. Mathar_, Jul 04 2013
%H Vincenzo Librandi, <a href="/A226592/b226592.txt">Table of n, a(n) for n = 0..2000</a>
%H V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H <a href="/index/Do#Dyck">Index to sequences related to dying rabbits</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,2,0,0,2,0,0,2).
%F For 0 <= n <= 4, a(n)=1;
%F for 5 <= n <= 11, a(n) = a(n-3) + 2*a(n-5);
%F for 12 <= n, a(n) = 2*( a(n-5) + a(n-8) + a(n-11) ).
%F G.f: 1-x*(1 +x +x^2 +x^3 +3*x^4 +x^5 +x^6 +3*x^7 +x^8 +x^9 +3*x^10) / ( -1 +2*x^5 +2*x^8 +2*x^11 ). - _R. J. Mathar_, Jul 04 2013
%t CoefficientList[Series[1 - x (1 + x + x^2 + x^3 + 3 x^4 + x^5 + x^6 + 3 x^7 + x^8 + x^9 + 3 x^10)/(-1 + 2 x^5 + 2 x^8 + 2 x^11), {x, 0, 60}], x] (* _Vincenzo Librandi_, Feb 17 2017 *)
%o (Magma) I:=[1,1,1,1,1,3,3,3,5,5,9,11]; [n le 12 select I[n] else 2*Self(n-5)+2*Self(n-8)+2*Self(n-11): n in [1..50]]; // _Vincenzo Librandi_, Feb 17 2017
%K nonn,easy,less
%O 0,6
%A _Lin Yin-Chen_, Jun 13 2013
%E New name from _Joerg Arndt_, Dec 11 2021