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%I #79 Dec 28 2022 01:56:55
%S 0,1,1,2,3,5,7,11,16,24,35,52,76,112,164,241,353,518,759,1113,1631,
%T 2391,3504,5136,7527,11032,16168,23696,34728,50897,74593,109322,
%U 160219,234813,344135,504355,739168,1083304,1587659,2326828,3410132,4997792,7324620,10734753
%N Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-5).
%C Diagonal sums of Riordan array (1/(1-x), x(1+x+x^2)) yield a(n+1). - _Paul Barry_, Feb 16 2005
%C The Ca2 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 lead to this sequence. - _Johannes W. Meijer_, Jul 20 2011
%C Number of ordered partitions of (n-1) into parts less than or equal to 3, where the order of the 2's is unimportant. (see example). - _David Neil McGrath_, Apr 26 2015
%C Number of ordered partitions of (n-1) into parts less than or equal to 4, where the order of the 1's is unimportant.(see example). - _David Neil McGrath_, May 05 2015
%C List the partitions of n in nonincreasing order. Freeze the 1's and 2's in place and allow the other summands to vary their order without disturbing the 1's and 2's. The result is a(n+1). - _Gregory L. Simay_ (based on correspondence with George E. Andrews), Jul 11 2016
%C Number of ordered partitions of n-1 where the order of the 1's and the 2's are unimportant. - _Gregory L. Simay_, Jul 18 2016
%H Michael De Vlieger, <a href="/A023435/b023435.txt">Table of n, a(n) for n = 0..6024</a>
%H J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H V. E. Hoggatt, Jr. and D. A. Lind, <a href="https://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H Z. Kasa, <a href="http://arxiv.org/abs/1104.4425">On scattered subword complexity</a>, arXiv:1104.4425 [cs.DM], 2011.
%H Anthony Shannon, François Dubeau, Mine Uysal, and Engin Özkan, <a href="https://doi.org/10.7546/ijba.2022.26.4.000899">A Difference Equation Model of Infectious Disease</a>, Int. J. Bioautomation (2022) Vol. 26, No. 4, 339-352.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,-1)
%F G.f.: x / ( (x-1)*(1+x)*(x^3+x-1) ). - _R. J. Mathar_, Nov 28 2011
%e There are 11 partitions of 6 into parts less than or equal to 3, where the order of 2's is unimportant, a(7)=11. These are (33),(321=231=312),(132=123=213),(3111),(1311),(1131),(1113),(222),(2211=1122=1221=2112=2121=1212),(21111=12111=11211=11121=11112),(111111). - _David Neil McGrath_, Apr 26 2015
%e There are 11 partitions of 6 into parts less than equal to 4, where the order of 1's is unimportant. These are (42),(24),(411=141=114),(33),(321=312=132),(231=213=123),(3111=1311=1131=1113),(222),(2211=1122=2112=1221=1212=2121),(21111=12111=11211=11121=11112),(111111). - _David Neil McGrath_, May 05 2015
%e There are a(9)=24 partitions of 8 where the 1's and 2's are frozen []: (8), (7[1]), (6[2]), (53), (35) (44), (6[1][1]), (5,[2][1]), (43[1]), (34[1]), (4[2][2]), (33[2][2]) (5[1][1][1]), (4[2][1][1]), (33[1][1]), (3[2][2][1]), ([2][2][2][2]), (4[1][1][1][1]), (3[2][1][1][1]), ([2][2][2][1][1]), (3[1][1][1][1][1]), ([2][2][1][1][1][1]), ([2][1][1][1][1][1][1]),([1][1][1][1][1][1][1][1]). - _Gregory L. Simay_, Jul 11 2016
%t LinearRecurrence[{1, 1, 0, 0, -1}, {0, 1, 1, 2, 3}, 50] (* _Vincenzo Librandi_, Apr 27 2015 *)
%o (Magma) I:=[0,1,1,2,3]; [n le 5 select I[n] else Self(n-1)+Self(n-2)-Self(n-5): n in [1..45]]; // _Vincenzo Librandi_, Apr 27 2015
%o (PARI) x='x+O('x^99); concat(0, Vec(x/((x-1)*(1+x)*(x^3+x-1)))) \\ _Altug Alkan_, Apr 09 2018
%Y First differences are in A013979.
%Y Cf. A077864 (bisection).
%K nonn
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Vincenzo Librandi_, Apr 27 2015
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