A023435 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-5).

0, 1, 1, 2, 3, 5, 7, 11, 16, 24, 35, 52, 76, 112, 164, 241, 353, 518, 759, 1113, 1631, 2391, 3504, 5136, 7527, 11032, 16168, 23696, 34728, 50897, 74593, 109322, 160219, 234813, 344135, 504355, 739168, 1083304, 1587659, 2326828, 3410132, 4997792, 7324620, 10734753

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Diagonal sums of Riordan array (1/(1-x), x(1+x+x^2)) yield a(n+1). - Paul Barry, Feb 16 2005

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

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

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

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

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

Examples

			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
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
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
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..6024
• John H. E. Cohn, Letter to the editor, Fib. Quart. 2 (1964), 108.
• Verner E. Hoggatt, Jr. and Douglas A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.
• Zoltán Kása, On scattered subword complexity, arXiv:1104.4425 [cs.DM], 2011.
• William J. Keith, Robert Schneider, and Andrew V. Sills, Composition-theoretic series and false theta functions, Integers (2024) Vol. 24A, Art. No. A11. See p. 11.
• Anthony Shannon, François Dubeau, Mine Uysal, and Engin Özkan, A Difference Equation Model of Infectious Disease, Int. J. Bioautomation (2022) Vol. 26, No. 4, 339-352.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1).

Crossrefs

First differences are in A013979.

Cf. A077864 (bisection).

Programs

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

Mathematica

LinearRecurrence[{1, 1, 0, 0, -1}, {0, 1, 1, 2, 3}, 50] (* Vincenzo Librandi, Apr 27 2015 *)

PARI

x='x+O('x^99); concat(0, Vec(x/((x-1)*(1+x)*(x^3+x-1)))) \\ Altug Alkan, Apr 09 2018

Formula

G.f.: x / ( (x-1)*(1+x)*(x^3+x-1) ). - R. J. Mathar, Nov 28 2011

Extensions

More terms from Vincenzo Librandi, Apr 27 2015