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A023434 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4). 11
0, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 113, 150, 199, 264, 350, 464, 615, 815, 1080, 1431, 1896, 2512, 3328, 4409, 5841, 7738, 10251, 13580, 17990, 23832, 31571, 41823, 55404, 73395, 97228, 128800, 170624, 226029, 299425, 396654, 525455 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

lim n->inf a(n)/a(n-1) = positive root of 1+x-x^3 (smallest Pisot-Vijayaraghavan number, A060006). - Gerald McGarvey, Sep 19 2004

a(n) = number of distinct even run-types taken over nonempty subsets of [n+1]. The run-type of a set of positive integers is the sequence of lengths when the set is decomposed into maximal runs of consecutive integers and it is even if all its entries are even. For example, the set {2,3,5,6,9,10,11} has run-type (2,2,3) and a(6)=6 counts (2),(4),(6),(2,2),(2,4),(4,2). - David Callan, Jul 14 2006

Partial sums of the sequence obtained by deleting the first 2 terms of A000931. Example: 0+1+0+1+1 = 3 = a(4). - David Callan, Jul 14 2006

One less than the sequence obtained by deleting the first 7 terms of A000931. - Ira M. Gessel, May 02 2007

This sequence counts ordered partitions of (n-1) into parts less than or equal to 3, in which the order of 1's are unimportant. Alternately, the order of 2's and 3's are important (see example). - David Neil McGrath, Apr 26 2015

LINKS

Robert Israel, Table of n, a(n) for n = 0..7360

O. Bouillot, The Algebra of Multitangent Functions, 2013.

J. H. E. Cohn, Letter to the editor, Fib. Quart. 2 (1964), 108.

V. E. Hoggatt, Jr. and D. A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1070

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

FORMULA

a(n) = A000931(n+7)-1.

a(0)=0, a(1)=1, a(2)=1 then for n>2 a(n)=ceiling(r*a(n-1)) where r is the positive root of x^3-x-1=0. - Benoit Cloitre, Jun 19 2004

G.f.: x/((1-x)(1-x^2-x^3)). - Jon Perry, Jul 04 2004

For n>2 a(n) = floor(sqrt(a(n-3)*a(n-2) + a(n-2)*a(n-1) + a(n-1)*a(n-3))) + 1. - Gerald McGarvey, Sep 19 2004

a(n) = Sum_{1<=k<=(n+2)/3} binomial[Floor[(n+2-k)/2],k]. This formula counts even run-types by length. - David Callan, Jul 14 2006

a(n) = a(n-2) + a(n-3) + 1. - M. Dols (markdols(AT)yahoo.com), Feb 01 2010

a(n) + a(n+1) = A054405(n). Partial sums is A054405. - Michael Somos, Dec 01 2013

a(-3-n) = -A077905(n) for all n in Z. - Michael Somos, Sep 25 2014

EXAMPLE

G.f. = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 11*x^8 + ...

a(7)=8, with (n-1)=6. The partially ordered partitions of 6 are (33),(321,312,132=one),(231,213,123=one),(3111,1311,1131,1113=one),(222),(2211,1122,1221,2112,1212,2121=one),(21111,12111,11211,11121,11112=one),(111111). - David Neil McGrath, Apr 26 2015

MAPLE

f:= gfun:-rectoproc({a(n)=a(n-1)+a(n-2)-a(n-4), seq(a(i)=[0, 1, 1, 2][i+1], i=0..3)}, a(n), remember):

seq(f(i), i=0..100); # Robert Israel, May 04 2015

MATHEMATICA

a=b=c=0; d=1; lst={c, d}; Do[e=c+d-a; AppendTo[lst, e]; a=b; b=c; c=d; d=e, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 30 2009 *)

a[ n_] := If[ n < 0, SeriesCoefficient[ -x^3 / (1 - x^2 - x^3 + x^4), {x, 0, -n}], SeriesCoefficient[ x / (1 - x - x^2 + x^4), {x, 0, n}]]; (* Michael Somos, Nov 29 2013 *)

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

PROG

(PARI) {a(n) = polcoeff( if( n<0, -x^3 / (1 - x^2 - x^3 + x^4), x / (1 - x - x^2 + x^4)) + x * O(x^abs(n)), abs(n))}; /* Michael Somos, Nov 29 2013 */

(MAGMA) [0, 1] cat [ n le 4 select (n) else Self(n-1)+Self(n-2)-Self(n-4): n in [1..45] ]; // Vincenzo Librandi, Apr 27 2015

CROSSREFS

Cf. A000931, A054405, A060006, A077905.

Sequence in context: A036001 A027336 A237830 * A087192 A188917 A046935

Adjacent sequences:  A023431 A023432 A023433 * A023435 A023436 A023437

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 23 23:53 EDT 2017. Contains 286937 sequences.