OFFSET
0,5
COMMENTS
The dying rabbits problem of immortal rabbits and matureness after 1 month defines the Fibonacci sequence.
For 0 <= n <= 9, a(n) = A003269(n+1), but a(10) = A003269(11) - 1 because of the death of the first pair of rabbits. - Robert FERREOL, Oct 05 2017
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Antonio M. Oller-Marcén, The Dying Rabbit Problem Revisited, INTEGERS 9 (2009), 129-138
Index entries for linear recurrences with constant coefficients, signature (1, -1, 1, 0, 1, 0, 1, 0, 1).
FORMULA
G.f.: -(1 + x + x^2 + x^3 + x^4)*(x^4 - x^3 + x^2 - x + 1)/(-1 + x - x^2 + x^3 + x^5 + x^7 + x^9). - R. J. Mathar, May 12 2009
G.f.: (1 - x^10) / (1 - x - x^4 + x^11) = 1 / (1 - x / (1 - x^3 / (1 + x^3 / (1 - x^3 / (1 + x^3 / (1 - x / (1 + x / (1 - x / (1 + x))))))))). - Michael Somos, Jan 03 2013
a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-5) + a(n-7) + a(n-9). - Joerg Arndt, Oct 04 2017
a(n)=1 for 0 <= n <= 3, a(n) = a(n-1) + a(n-4) for 4 <= n <= 9, and a(n) = a(n-4) + a(n-5) + ... + a(n-10) for n >= 10. - Robert FERREOL, Oct 04 2017
EXAMPLE
The number of pairs at the 13th month is 32.
MAPLE
Cnh := proc(n, h) option remember ; if n < 0 then 0 ; elif n < h then 1; else procname(n-1, h)+procname(n-h, h) ; fi; end:
C := proc(n, k, h) option remember ; local i; if n >= 0 and n < k+h-1 then Cnh(n, h); else add( procname(n-h-i, k, h), i=0..k-1) ; fi; end:
MATHEMATICA
LinearRecurrence[{1, -1, 1, 0, 1, 0, 1, 0, 1}, {1, 1, 1, 1, 2, 3, 4, 5, 7}, 50] (* Harvey P. Dale, Apr 23 2011 *)
PROG
(PARI) {a(n) = if( n<0, n = -n; polcoeff( (x^6 - x^10) / (1 - x^7 - x^10 + x^11) + x * O(x^n), n), polcoeff( (1 - x^10) / (1 - x - x^4 + x^11) + x * O(x^n), n))} /* Michael Somos, Jan 03 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Parthasarathy Nambi, May 09 2009
EXTENSIONS
Edited and extended by R. J. Mathar, May 12 2009
Name corrected by Robert FERREOL, Nov 18 2017
STATUS
approved