A077864 -id:A077864 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

			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

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).

First differences are in A013979.

Cf. A077864 (bisection).

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

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

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

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

More terms from Vincenzo Librandi, Apr 27 2015

A124234 Riordan array (1/(1-x), x(1+x)^2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 4, 11, 7, 1, 1, 4, 15, 22, 9, 1, 1, 4, 16, 42, 37, 11, 1, 1, 4, 16, 57, 93, 56, 13, 1, 1, 4, 16, 63, 163, 176, 79, 15, 1, 1, 4, 16, 64, 219, 386, 299, 106, 17, 1, 1, 4, 16, 64, 247, 638, 794, 470, 137, 19, 1

Offset: 0

Paul Barry, Oct 22 2006

Row sums are A077864. Diagonal sums are A004695(n+3). T(2n,n) is A032443.

			Triangle begins
1,
1, 1,
1, 3, 1,
1, 4, 5, 1,
1, 4, 11, 7, 1,
1, 4, 15, 22, 9, 1,
1, 4, 16, 42, 37, 11, 1

Cf. A004695, A032443, A077864.

tabl(nn) = for (n=0, nn, for (k=0, n, print1(sum(j=0, n-k, binomial(2*k, j)), ", ")); print();); \\ Michel Marcus, Nov 05 2016

T(n,k) = Sum_{j=0..n-k} C(2k,j).

A192805 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+2x+1. See Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 25, 53, 113, 242, 519, 1114, 2392, 5137, 11033, 23697, 50898, 109323, 234814, 504356, 1083305, 2326829, 4997793, 10734754, 23057167, 49524466, 106373552, 228479649, 490751217, 1054084065, 2264066146, 4862985491

Offset: 0

Clark Kimberling, Jul 10 2011

nonn

For discussions of polynomial reduction, see A192232 and A192744.

			The first five polynomials p(n,x) and their reductions:
p(1,x)=1 -> 1
p(2,x)=x+1 -> x+1
p(3,x)=x^2+x+1 -> x^2+x+1
p(4,x)=x^3+x^2+x+1 -> 2x^2+3x+2
p(5,x)=x^4+x^3+x^2+x+1 -> 5x^2+6*x+3, so that
A192805=(1,1,1,2,3,...), A002478=(0,1,1,3,6,...), A077864=(0,0,1,2,5,...).

Index entries for linear recurrences with constant coefficients, signature (2,1,-1,-1).

Cf. A192744, A192232, A002478.

q = x^3; s = x^2 + 2 x + 1; z = 40;
p[0, x_] := 1; p[n_, x_] := x^n + p[n - 1, x];
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192805 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A002478  *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]
  (* A077864 *)

a(n)=2*a(n-1)+a(n-2)-a(n-3)-a(n-4).
G.f.: -(1+x)*(2*x-1) / ( (x-1)*(x^3+2*x^2+x-1) ). - R. J. Mathar, May 06 2014
a(n)-a(n-1) = A002478(n-3). - R. J. Mathar, May 06 2014

A262735 Expansion of x(1-x-x^2)/((1-x)(1-x-2*x^2-x^3)).

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 17, 36, 77, 165, 354, 760, 1632, 3505, 7528, 16169, 34729, 74594, 160220, 344136, 739169, 1587660, 3410133, 7324621, 15732546, 33791920, 72581632, 155898017, 334853200, 719230865, 1544835281, 3318150210, 7127051636, 15308187336

Offset: 0

Werner Schulte, Sep 29 2015

Index entries for linear recurrences with constant coefficients, signature (2,1,-1,-1).

Cf. A027907, A077864.

a:=proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 1 elif n=3 then 2 else 2*a(n-1)+a(n-2)-a(n-3)-a(n-4); fi; end:  seq(f(n), n=0..50); # Wesley Ivan Hurt, Oct 10 2015

CoefficientList[Series[x (1 - x - x^2)/((1 - x) (1 - x - 2 x^2 - x^3)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 29 2015 *)
LinearRecurrence[{2,1,-1,-1},{0,1,1,2},40] (* Harvey P. Dale, Sep 23 2019 *)

concat(0, Vec(x*(1-x-x^2)/((1-x)*(1-x-2*x^2-x^3)) + O(x^50))) \\ Michel Marcus, Sep 29 2015

G.f.: A(x) = x*(1-x-x^2)*B(x), where B is g.f. of A077864.
a(n) = A077864(n+1)-2*A077864(n), n >= 0.
a(n+3) = A077864(n+2)-A077864(n+1)-A077864(n), n >= 0.
Recurrence: a(0)=0, a(1)=1, a(2)=1, a(3)=2, a(4)=4, a(5)=8, a(6)=17, and a(n) = 4*a(n-1)-4*a(n-2)+a(n-7) for n >= 7.
Conjecture: a(n+1) = Sum_{j=0..n/2} A027907(n-j,2*j), n >= 0.
a(n) = 2*a(n-1)+a(n-2)-a(n-3)-a(n-4) for n>3. - Wesley Ivan Hurt, Oct 10 2015
a(n) = a(n-1)+2*a(n-2)+a(n-3)-1, n>=3. - R. J. Mathar, Nov 07 2015

cp's OEIS Frontend

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

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A124234 Riordan array (1/(1-x), x(1+x)^2).

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A192805 Constant term in the reduction of the polynomial 1+x+x^2+...+x^n by x^3->x^2+2x+1. See Comments.

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A262735 Expansion of x*(1-x-x^2)/((1-x)*(1-x-2*x^2-x^3)).

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A262735 Expansion of x(1-x-x^2)/((1-x)(1-x-2*x^2-x^3)).