

A156040


Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.


19



1, 1, 3, 4, 6, 8, 11, 13, 17, 20, 24, 28, 33, 37, 43, 48, 54, 60, 67, 73, 81, 88, 96, 104, 113, 121, 131, 140, 150, 160, 171, 181, 193, 204, 216, 228, 241, 253, 267, 280, 294, 308, 323, 337, 353, 368, 384, 400, 417, 433, 451, 468, 486, 504, 523, 541, 561, 580, 600
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OFFSET

0,3


COMMENTS

For n = 1, 2 these are just the triangular numbers. a(n) is always at least 1/3 of the corresponding triangular number, since each partition of this type gives up to three ordered partitions with the same cyclical order.
An alternative definition, which avoids using parts of size 0: a(n) is the third diagonal of A184957.  N. J. A. Sloane, Feb 27 2011
Diagonal sums of the triangle formed by rows T(2, k) k = 0, 1, ..., 2m of ascending mnomial triangles (see A004737):
1
1 2 1
1 2 3 2 1
1 2 3 4 3 2 1
1 2 3 4 5 4 3 2 1
1 2 3 4 5 6 5 4 3 2 1
 Bob Selcoe, Feb 07 2014
Arrange A004396 in rows successively shifted to the right two spaces and sum the columns:
1 1 2 3 3 4 5 5 6 ...
1 1 2 3 3 4 5 ...
1 1 2 3 3 ...
1 1 2 ...
1 ...

1 1 3 4 6 8 11 13 17 ...  L. Edson Jeffery, Jul 30 2014
a(n) is the dimension of threedimensional (2n + 2)homogeneous polynomial vector fields with full tetrahedral symmetry (for a given orthogonal representation), and which are solenoidal.  Giedrius Alkauskas, Sep 30 2017
Also the number of compositions of n + 3 into three parts, the first at least as great as each of the other two. Also the number of compositions of n + 4 into three parts, the first strictly greater than each of the other two.  Gus Wiseman, Oct 09 2020


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG] (2017), Section 4.3.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,1,1).


FORMULA

G.f.: (x^2+1) / (1xx^2+x^4+x^5x^6).  Alois P. Heinz, Jun 14 2009
Slightly nicer g.f.: (1+x^2)/((1x)*(1x^2)*(1x^3)).  N. J. A. Sloane, Apr 29 2011
a(n) = A007590(n+2)  A000212(n+2).  Richard R. Forberg, Dec 08 2013
a(2*n) = A071619(n+1).  L. Edson Jeffery, Jul 29 2014
a(n) = a(n1) + a(n2)  a(n4)  a(n5) + a(n6), with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 6, a(5) = 8.  Harvey P. Dale, May 28 2015
a(n) = (n^2 + 4*n + 3)/6 + IF(MOD(n, 2) = 0, 1/2) + IF(MOD(n, 3) = 1, 1/3).  Heinrich Ludwig, Mar 21 2017
a(n) = 1 + floor((n^2 + 4*n)/6).  Giovanni Resta, Mar 21 2017
Euler transform of length 4 sequence [1, 2, 1, 1].  Michael Somos, Mar 26 2017
a(n) = a(4  n) for all n in Z.  Michael Somos, Mar 26 2017
0 = a(n)*(1 + a(n)  2*a(n+1)  2*a(n+2) + 2*a(n+3)) + a(n+1)*(+1 + a(n+1) + 2*a(n+2)  2*a(n+3)) + a(n+2)*(+1 + a(n+2)  2*a(n+3)) + a(n+3)*(1 + a(n+3)) for all n in Z.  Michael Somos, Mar 26 2017
a(n) = round((n+1)*(n+3)/6).  Bill McEachen, Feb 16 2021


EXAMPLE

G.f. = 1 + x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + 20*x^9 + ...
The a(4) = 6 compositions of 4 are: (4 0 0), (3 1 0), (3 0 1), (2 2 0), (2 1 1), (2 0 2).
From Gus Wiseman, Oct 05 2020: (Start)
The a(0) = 1 through a(7) = 13 triples of nonnegative integers summing to n where the first is at least as great as each of the other two are:
(000) (100) (101) (111) (202) (212) (222) (313)
(110) (201) (211) (221) (303) (322)
(200) (210) (220) (302) (312) (331)
(300) (301) (311) (321) (403)
(310) (320) (330) (412)
(400) (401) (402) (421)
(410) (411) (430)
(500) (420) (502)
(501) (511)
(510) (520)
(600) (601)
(610)
(700)
(End)


MAPLE

a:= proc(n) local m, r; m := iquo(n, 6, 'r'); (4 +6*m +2*r) *m + [1, 1, 3, 4, 6, 8][r+1] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009


MATHEMATICA

nn = 58; CoefficientList[Series[x^3/(1  x^2)^2/(1  x^3) + 1/(1  x^2)^2/(1  x), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 14 2013 *)
CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1  x)^3), {x, 0, 58}], x] (* L. Edson Jeffery, Jul 29 2014 *)
LinearRecurrence[{1, 1, 0, 1, 1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3, {3}], #[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]], {n, 0, 15}] (* Gus Wiseman, Oct 05 2020*)


PROG

(PARI) {a(n) = n*(n+4)\6 + 1}; /* Michael Somos, Mar 26 2017 */


CROSSREFS

For compositions into 4 summands see A156039; also see A156041 and A156042.
Cf. A184957, A071619 (bisection).
A001399(n2)*2 is the strict case.
A001840(n2) is the version with opposite relations.
A001840(n1) is the version with strict opposite relations.
A069905 is the case with strict relations.
A014311 ranks 3part compositions, with strict case A337453.
A014612 ranks 3part partitions, with strict case A007304.
Cf. A000217, A001523, A211540, A218004, A220377, A337483, A337484.
Sequence in context: A030761 A060903 A079401 * A325172 A242254 A107770
Adjacent sequences: A156037 A156038 A156039 * A156041 A156042 A156043


KEYWORD

nonn,easy


AUTHOR

Jack W Grahl, Feb 02 2009, Feb 11 2009


EXTENSIONS

More terms from Alois P. Heinz, Jun 14 2009


STATUS

approved



