A226592 a(n) = 2*( a(n-5) + a(n-8) + a(n-11) ) for n >= 12.

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, 272, 432, 472, 620, 876, 940, 1412, 1700, 2044, 3000, 3320, 4608, 6016, 6912, 10064, 11792, 15184, 20856, 23864, 33432, 41616, 50832, 71056, 83344, 111056, 145072, 172976

Offset

0,6

Comments

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.

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)*(1-2*x^5-2*x^8-2*x^11)) = (x+1) *(x^2+1) *(x^2+x+1) *(x^2-x+1) *(x^4-x^2+1) / (1-2*x^5-2*x^8-2*x^11). - R. J. Mathar, Jul 04 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

V. E. Hoggatt, Jr. and D. A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.

Index to sequences related to dying rabbits

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,2,0,0,2).

Formula

For 0 <= n <= 4, a(n)=1;

for 5 <= n <= 11, a(n) = a(n-3) + 2*a(n-5);

for 12 <= n, a(n) = 2*( a(n-5) + a(n-8) + a(n-11) ).

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

Mathematica

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 *)

Prog

(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

Crossrefs

Sequence in context: A096015 A046702 A357057 * A133683 A182998 A117900

Adjacent sequences: A226589 A226590 A226591 * A226593 A226594 A226595

Keyword

nonn,easy,less

Author

Lin Yin-Chen, Jun 13 2013

Extensions

New name from Joerg Arndt, Dec 11 2021

Status

approved