%I #107 Apr 05 2023 18:50:50

%S 1,1,3,4,6,8,11,13,17,20,24,28,33,37,43,48,54,60,67,73,81,88,96,104,

%T 113,121,131,140,150,160,171,181,193,204,216,228,241,253,267,280,294,

%U 308,323,337,353,368,384,400,417,433,451,468,486,504,523,541,561,580,600

%N 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.

%C 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.

%C 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

%C Diagonal sums of the triangle formed by rows T(2, k) k = 0, 1, ..., 2m of ascending m-nomial triangles (see A004737):

%C 1

%C 1 2 1

%C 1 2 3 2 1

%C 1 2 3 4 3 2 1

%C 1 2 3 4 5 4 3 2 1

%C 1 2 3 4 5 6 5 4 3 2 1

%C - _Bob Selcoe_, Feb 07 2014

%C Arrange A004396 in rows successively shifted to the right two spaces and sum the columns:

%C 1 1 2 3 3 4 5 5 6 ...

%C 1 1 2 3 3 4 5 ...

%C 1 1 2 3 3 ...

%C 1 1 2 ...

%C 1 ...

%C ------------------------------

%C 1 1 3 4 6 8 11 13 17 ... - _L. Edson Jeffery_, Jul 30 2014

%C a(n) is the dimension of three-dimensional (2n + 2)-homogeneous polynomial vector fields with full tetrahedral symmetry (for a given orthogonal representation), and which are solenoidal. - _Giedrius Alkauskas_, Sep 30 2017

%C 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

%H Vincenzo Librandi, <a href="/A156040/b156040.txt">Table of n, a(n) for n = 0..1000</a>

%H Giedrius Alkauskas, <a href="https://arxiv.org/abs/1601.06570">Projective and polynomial superflows. I</a>, arxiv.org/1601.06570 [math.AG] (2017), Section 4.3.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).

%F G.f.: (x^2+1) / (1-x-x^2+x^4+x^5-x^6). - _Alois P. Heinz_, Jun 14 2009

%F Slightly nicer g.f.: (1+x^2)/((1-x)*(1-x^2)*(1-x^3)). - _N. J. A. Sloane_, Apr 29 2011

%F a(n) = A007590(n+2) - A000212(n+2). - _Richard R. Forberg_, Dec 08 2013

%F a(2*n) = A071619(n+1). - _L. Edson Jeffery_, Jul 29 2014

%F a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), 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

%F 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

%F a(n) = 1 + floor((n^2 + 4*n)/6). - _Giovanni Resta_, Mar 21 2017

%F Euler transform of length 4 sequence [1, 2, 1, -1]. - _Michael Somos_, Mar 26 2017

%F a(n) = a(-4 - n) for all n in Z. - _Michael Somos_, Mar 26 2017

%F 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

%F a(n) = round((n+1)*(n+3)/6). - _Bill McEachen_, Feb 16 2021

%F Sum_{n>=0} 1/a(n) = 3/2 + Pi^2/36 + (tan(c1)-1)*c1 + 3*c2*sinh(c2)/(1+2*cosh(c2)), where c1 = Pi/(2*sqrt(3)) and c2 = Pi*sqrt(2)/3. - _Amiram Eldar_, Dec 10 2022

%F E.g.f.: ((16 + 15*x + 3*x^2)*cosh(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (7 + 15*x + 3*x^2)*sinh(x))/18. - _Stefano Spezia_, Apr 05 2023

%e 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 + ...

%e 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).

%e From _Gus Wiseman_, Oct 05 2020: (Start)

%e 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:

%e (000) (100) (101) (111) (202) (212) (222) (313)

%e (110) (201) (211) (221) (303) (322)

%e (200) (210) (220) (302) (312) (331)

%e (300) (301) (311) (321) (403)

%e (310) (320) (330) (412)

%e (400) (401) (402) (421)

%e (410) (411) (430)

%e (500) (420) (502)

%e (501) (511)

%e (510) (520)

%e (600) (601)

%e (610)

%e (700)

%e (End)

%p 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

%t 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 *)

%t CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1 - x)^3), {x, 0, 58}], x] (* _L. Edson Jeffery_, Jul 29 2014 *)

%t LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* _Harvey P. Dale_, May 28 2015 *)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3,{3}],#[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]],{n,0,15}] (* _Gus Wiseman_, Oct 05 2020*)

%o (PARI) {a(n) = n*(n+4)\6 + 1}; /* _Michael Somos_, Mar 26 2017 */

%Y For compositions into 4 summands see A156039; also see A156041 and A156042.

%Y Cf. A184957, A071619 (bisection).

%Y A001399(n-2)*2 is the strict case.

%Y A001840(n-2) is the version with opposite relations.

%Y A001840(n-1) is the version with strict opposite relations.

%Y A069905 is the case with strict relations.

%Y A014311 ranks 3-part compositions, with strict case A337453.

%Y A014612 ranks 3-part partitions, with strict case A007304.

%Y Cf. A000217, A001523, A211540, A218004, A220377, A337483, A337484.

%K nonn,easy

%O 0,3

%A _Jack W Grahl_, Feb 02 2009, Feb 11 2009

%E More terms from _Alois P. Heinz_, Jun 14 2009