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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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 m-nomial 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 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

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) / (1-x-x^2+x^4+x^5-x^6). - Alois P. Heinz, Jun 14 2009

Slightly nicer g.f.: (1+x^2)/((1-x)*(1-x^2)*(1-x^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(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

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

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).

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 *)

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).

Sequence in context: A030711 A060903 A079401 * A242254 A107770 A067054

Adjacent sequences:  A156037 A156038 A156039 * A156041 A156042 A156043

KEYWORD

nonn

AUTHOR

Jack W Grahl, Feb 02 2009, Feb 11 2009

EXTENSIONS

More terms from Alois P. Heinz, Jun 14 2009

STATUS

approved

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Last modified February 25 13:30 EST 2018. Contains 299654 sequences. (Running on oeis4.)