A289692 -id:A289692 - cp's OEIS frontend

A023434 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4).

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

Offset: 0

N. J. A. Sloane

Limit_{n->infinity} a(n)/a(n-1) = positive root of 1+x-x^3 (smallest Pisot-Vijayaraghavan number, A060006). - Gerald McGarvey, Sep 19 2004
a(n) is the 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
Interleaving of A289692 and A077855. - Bruce J. Nicholson, Apr 09 2018

			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

Robert Israel, Table of n, a(n) for n = 0..7360
O. Bouillot, The Algebra of Multitangent Functions, arXiv:1404.0992 [math.NT], 2014.
O. Bouillot, The Algebra of Multitangent Functions, Journal of Algebra, Volume 410, 15 July 2014, Pages 148-238.
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).

Cf. A000931, A054405, A060006, A077905.

Cf. A289692, A077855.

[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

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

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 *)

{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 */

x='x+O('x^99); concat(0, Vec(x/((1-x)*(1-x^2-x^3)))) \\ Altug Alkan, Apr 09 2018

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_{k=1..floor((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. - Mark Dols, 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

A289693 The number of partitions of [n] with exactly 3 blocks without peaks.

Original entry on oeis.org

0, 0, 1, 3, 9, 27, 75, 197, 503, 1263, 3132, 7695, 18784, 45649, 110585, 267276, 644907, 1554208, 3742321, 9005265, 21659603, 52078400, 125186565, 300870586, 723010749, 1737273406, 4174084259, 10028409724, 24092769583, 57880137331

Offset: 1

R. J. Mathar, Jul 09 2017

nonn

T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Table 1.
Index entries for linear recurrences with constant coefficients, signature (6,-15,24,-29,25,-17,9,-3,1).

Cf. A289692, A289694.

with(orthopoly) :
nmax := 10:
tpr := 1+x^2/2 :
k := 3:
g := x^k ;
for j from 1 to k do
        if j> 1 then
                g := g*( U(j-1,tpr)-(1+x)*U(j-2,tpr)) / ((1-x)*U(j-1,tpr)-U(j-2,tpr)) ;
        else
                # note that U(-1,.)=0, U(0,.)=1
                g := g* U(j-1,tpr) / ((1-x)*U(j-1,tpr)) ;
        end if;
end do:
simplify(%) ;
taylor(g,x=0,nmax+1) ;
gfun[seriestolist](%) ; # R. J. Mathar, Mar 11 2021

From Colin Barker, Nov 07 2017: (Start)
G.f.: x^3*(1 - x + x^2)*(1 - 2*x + 3*x^2 - x^3 + x^4) / ((1 - x)*(1 - 2*x + x^2 - x^3)*(1 - 3*x + 3*x^2 - 4*x^3 + x^4 - x^5)).
a(n) = 6*a(n-1) - 15*a(n-2) + 24*a(n-3) - 29*a(n-4) + 25*a(n-5) - 17*a(n-6) + 9*a(n-7) - 3*a(n-8) + a(n-9) for n>9.
(End)

A289694 The number of partitions of [n] with exactly 4 blocks without peaks.

Original entry on oeis.org

0, 0, 0, 1, 4, 16, 64, 236, 818, 2736, 8934, 28622, 90324, 281792, 871556, 2677750, 8184383, 24913238, 75593383, 228793147, 691094857, 2084237036, 6277871658, 18890568921, 56798001639, 170665733660, 512554832309, 1538718547049

Offset: 1

R. J. Mathar, Jul 09 2017

T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Table 1.
Index entries for linear recurrences with constant coefficients, signature (10,-45,130,-280,471,-643,734,-701,575,-400,237,-121,49,-18,4,-1).

Cf. A289692 (2 blocks), A289693 (3 blocks).

with(orthopoly) :
nmax := 15:
tpr := 1+x^2/2 :
k := 4:
g := x^k ;
for j from 1 to k do
    if j> 1 then
        g := g*( U(j-1,tpr)-(1+x)*U(j-2,tpr)) / ((1-x)*U(j-1,tpr)-U(j-2,tpr)) ;
    else
        # note that U(-1,.)=0, U(0,.)=1
        g := g* U(j-1,tpr) / ((1-x)*U(j-1,tpr)) ;
    end if;
end do:
simplify(%) ;
taylor(g,x=0,nmax+1) ;
gfun[seriestolist](%) ; # R. J. Mathar, Mar 11 2021

G.f. x^4*(x^2-x+1)*(x^4-x^3+3*x^2-2*x+1)*(x^6-x^5+5*x^4-4*x^3+6*x^2-3*x+1) / ( (x-1)*(x^5-x^4+4*x^3-3*x^2+3*x-1)*(x^7-x^6+6*x^5-5*x^4+10*x^3-6*x^2+4*x-1)*(x^3-x^2+2*x-1) ). - R. J. Mathar, Mar 11 2021

a(n)= 10*a(n-1) -45*a(n-2) +130*a(n-3) -280*a(n-4) +471*a(n-5) -643*a(n-6) +734*a(n-7) -701*a(n-8) +575*a(n-9) -400*a(n-10) +237*a(n-11) -121*a(n-12) +49*a(n-13) -18*a(n-14) +4*a(n-15) -a(n-16). - R. J. Mathar, Mar 11 2021

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A023434 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4).

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A289693 The number of partitions of [n] with exactly 3 blocks without peaks.

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A289694 The number of partitions of [n] with exactly 4 blocks without peaks.

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