|
%I #48 Nov 29 2017 04:33:05
%S 1,1,1,1,2,3,4,5,7,10,13,17,23,32,44,59,79,107,146,198,267,361,490,
%T 665,900,1217,1648,2234,3027,4098,5548,7515,10181,13789,18672,25287,
%U 34251,46392,62830,85090,115243,156087,211402,286311,387765,525180,711295,963355,1304728
%N Number of pairs of rabbits in month n in the dying rabbits problem, if they become mature after 4 months and give birth to exactly 7 pairs, one per month.
%C The dying rabbits problem of immortal rabbits and matureness after 1 month defines the Fibonacci sequence.
%C 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
%H G. C. Greubel, <a href="/A160333/b160333.txt">Table of n, a(n) for n = 0..1000</a>
%H Antonio M. Oller-Marcén, <a href="http://www.emis.de/journals/INTEGERS/papers/j11/j11.Abstract.html">The Dying Rabbit Problem Revisited</a>, INTEGERS 9 (2009), 129-138
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, -1, 1, 0, 1, 0, 1, 0, 1).
%F 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
%F 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
%F 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
%F 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
%e The number of pairs at the 13th month is 32.
%p 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:
%p 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:
%p A160333 := proc(n) C(n,7,4) ; end: seq(A160333(n),n=0..80) ; # _R. J. Mathar_, May 12 2009
%t 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 *)
%o (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 */
%Y Cf. A000045, A000930.
%K nonn
%O 0,5
%A _Parthasarathy Nambi_, May 09 2009
%E Edited and extended by _R. J. Mathar_, May 12 2009
%E Name corrected by _Robert FERREOL_, Nov 18 2017
|
