A001399 -id:A001399 - cp's OEIS frontend

A055684 Number of different n-pointed stars.

0, 0, 1, 0, 2, 1, 2, 1, 4, 1, 5, 2, 3, 3, 7, 2, 8, 3, 5, 4, 10, 3, 9, 5, 8, 5, 13, 3, 14, 7, 9, 7, 11, 5, 17, 8, 11, 7, 19, 5, 20, 9, 11, 10, 22, 7, 20, 9, 15, 11, 25, 8, 19, 11, 17, 13, 28, 7, 29, 14, 17, 15, 23, 9, 32, 15, 21, 11, 34, 11, 35, 17, 19, 17, 29, 11

Offset: 3

Robert G. Wilson v, Jun 09 2000

Does not count rotations or reflections.
This is also the distinct ways of writing a number as the sum of two positive integers greater than one that are coprimes. - Lei Zhou, Mar 19 2014
Equivalently, a(n) is the number of relatively prime 2-part partitions of n without 1's. The Heinz numbers of these partitions are the intersection of A001358 (pairs), A005408 (no 1's), and A000837 (relatively prime) or A302696 (pairwise coprime). - Gus Wiseman, Oct 28 2020

			The first star has five points and is unique. The next is the seven pointed star and it comes in two varieties.
From _Gus Wiseman_, Oct 28 2020: (Start)
The a(5) = 1 through a(17) = 7 irreducible pairs > 1 (shown as fractions, empty column indicated by dot):
  2/3  .  2/5  3/5  2/7  3/7  2/9  5/7  2/11  3/11  2/13  3/13  2/15
          3/4       4/5       3/8       3/10  5/9   4/11  5/11  3/14
                              4/7       4/9         7/8   7/9   4/13
                              5/6       5/8                     5/12
                                        6/7                     6/11
                                                                7/10
                                                                8/9
(End)

Mark A. Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, New York, 1999, page 58.

Lei Zhou, Table of n, a(n) for n = 3..10002
Alexander Bogomolny, Polygons: formality and intuition.. Includes applet to draw star polygons.
Vi Hart, Doodling in Math Class: Stars, Video (2010).
Hugo Pfoertner, Star-shaped regular polygons up to n=25.
Eric Weisstein's World of Mathematics, Star Polygon

Cf. A023022.
Cf. A053669 smallest skip increment, A102302 skip increment of densest star polygon.
A055684*2 is the ordered version.
A082023 counts the complement (reducible pairs > 1).
A220377, A337563, and A338332 count triples instead of pairs.
A000837 counts relatively prime partitions, with strict case A078374.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 and A337485 count pairwise coprime partitions with no 1's.
A302698 counts relatively prime partitions with no 1's, with strict case A337452.
A327516 counts pairwise coprime partitions, with strict case A305713.
A337450 counts relatively prime compositions with no 1's, with strict case A337451.
Cf. A001399, A101268, A140106, A337461, A337462, A338333.

with(numtheory): A055684 := n->(phi(n)-2)/2; seq(A055684(n), n=3..100);

Table[(EulerPhi[n]-2)/2, {n, 3, 50}]
Table[Length[Select[IntegerPartitions[n,{2}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 28 2020 *)

a(n) = A023022(n) - 1.

a(n) + A082023(n) = A140106(n). - Gus Wiseman, Oct 28 2020

A337563 Number of pairwise coprime unordered triples of positive integers > 1 summing to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 7, 1, 7, 3, 9, 2, 15, 3, 13, 5, 17, 4, 29, 5, 20, 8, 28, 8, 42, 8, 31, 14, 42, 10, 59, 12, 45, 21, 52, 14, 77, 17, 68, 26, 69, 19, 101, 26, 84, 34, 86, 25, 138, 28, 95, 43, 111, 36, 161, 35, 118, 52, 151

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

Such partitions are necessarily strict.

The Heinz numbers of these partitions are the intersection of A005408 (no 1's), A014612 (triples), and A302696 (coprime).

			The a(10) = 1 through a(24) = 15 triples (empty columns indicated by dots, A..J = 10..19):
  532  .  543  .  743  753  754  .  765  B53  875  975  985  B75  987
          732     752       853     873       974  B73  B65  D73  B76
                            952     954       A73  D53  B74       B85
                            B32     972       B54       B83       B94
                                    B43       B72       B92       BA3
                                    B52       D43       D54       C75
                                    D32       D52       D72       D65
                                                        E53       D74
                                                        H32       D83
                                                                  D92
                                                                  F72
                                                                  G53
                                                                  H43
                                                                  H52
                                                                  J32

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A055684 is the version for pairs.
A220377 allows 1's, with non-strict version A307719.
A337485 counts these partitions of any length.
A337563*6 is the ordered version.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 counts pairwise coprime partitions with no 1's.
A078374 counts relatively prime strict partitions.
A200976 and A328673 count pairwise non-coprime partitions.
A302696 ranks pairwise coprime partitions.
A302698 counts relatively prime partitions with no 1's.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A337452 counts relatively prime strict partitions with no 1's.
Cf. A007304, A082024, A101268, A284825, A332004, A337451, A337461, A337462, A337561, A337599, A337601, A337605.

Table[Length[Select[IntegerPartitions[n,{3}],!MemberQ[#,1]&&CoprimeQ@@#&]],{n,0,30}]

A337604 Number of ordered triples of positive integers summing to n, any two of which have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 6, 0, 13, 0, 15, 7, 21, 0, 37, 0, 39, 16, 45, 0, 73, 6, 66, 28, 81, 0, 130, 6, 105, 46, 120, 21, 181, 6, 153, 67, 189, 12, 262, 6, 213, 118, 231, 12, 337, 21, 306, 121, 303, 12, 433, 57, 369, 154, 378, 18, 583, 30, 435, 217, 465

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

The first relatively prime triple (15,10,6) is counted under a(31).

			The a(6) = 1 through a(15) = 7 triples (empty columns indicated by dots, A = 10):
  222  .  224  333  226  .  228  .  22A  339
          242       244     246     248  366
          422       262     264     266  393
                    424     282     284  555
                    442     336     2A2  636
                    622     363     428  663
                            426     446  933
                            444     464
                            462     482
                            624     626
                            633     644
                            642     662
                            822     824
                                    842
                                    A22

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A014311 intersected with A337666 ranks these compositions.
A337667 counts these compositions of any length.
A335402 lists the positions of zeros.
A337461 is the coprime instead of non-coprime version.
A337599 is the unordered version, with strict case A337605.
A337605*6 is the strict version.
A000741 counts relatively prime 3-part compositions.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 count pairwise non-relatively prime partitions.
A307719 counts pairwise coprime 3-part partitions.
A318717 counts pairwise non-coprime strict partitions.
A333227 ranks pairwise coprime compositions.
Cf. A000217, A001399, A014612, A051424, A082024, A178472, A220377, A284825, A305713, A327516, A333228, A337561.

stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],stabQ[#,CoprimeQ]&]],{n,0,100}]

A000741 Number of compositions of n into 3 ordered relatively prime parts.

Original entry on oeis.org

0, 0, 1, 3, 6, 9, 15, 18, 27, 30, 45, 42, 66, 63, 84, 84, 120, 99, 153, 132, 174, 165, 231, 180, 270, 234, 297, 270, 378, 276, 435, 360, 450, 408, 540, 414, 630, 513, 636, 552, 780, 558, 861, 690, 828, 759, 1035, 744, 1113, 870, 1104, 972, 1326, 945, 1380, 1116, 1386, 1218

Offset: 1

N. J. A. Sloane

			From _Gus Wiseman_, Oct 14 2020: (Start)
The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,2,4)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,3,3)  (1,3,4)
                    (2,1,2)  (1,4,1)  (1,4,2)  (1,4,3)
                    (2,2,1)  (2,1,3)  (1,5,1)  (1,5,2)
                    (3,1,1)  (2,3,1)  (2,1,4)  (1,6,1)
                             (3,1,2)  (2,2,3)  (2,1,5)
                             (3,2,1)  (2,3,2)  (2,3,3)
                             (4,1,1)  (2,4,1)  (2,5,1)
                                      (3,1,3)  (3,1,4)
                                      (3,2,2)  (3,2,3)
                                      (3,3,1)  (3,3,2)
                                      (4,1,2)  (3,4,1)
                                      (4,2,1)  (4,1,3)
                                      (5,1,1)  (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)
(End)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.
N. J. A. Sloane, Transforms

A000010 is the length-2 version.
A000217(n-2) does not require relative primality.
A000740 counts these compositions of any length.
A000742 is the length-4 version.
A000837 counts relatively prime partitions.
A023023 is the unordered version.
A101271 is the strict case.
A101391 has this as column k = 3.
A284825*6 is the pairwise non-coprime case.
A291166 intersected with A014311 ranks these compositions.
A337461 is the pairwise coprime instead of relatively prime version.
A337603 counts length-3 compositions whose distinct parts are pairwise coprime.
A337604 is the pairwise non-coprime instead of relatively prime version.
Cf. A001399, A007997, A023022, A078374, A337450, A337451, A337602.

with(numtheory):
mobtr:= proc(p)
          proc(n) option remember;
            add(mobius(n/d)*p(d), d=divisors(n))
          end
        end:
A000217:= n-> n*(n+1)/2:
a:= mobtr(n-> A000217(n-2)):
seq(a(n), n=1..58);  # Alois P. Heinz, Feb 08 2011

mobtr[p_] := Module[{f}, f[n_] := f[n] = Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; f]; A000217[n_] := n*(n+1)/2; a = mobtr[A000217[#-2]&]; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Mar 12 2014, after Alois P. Heinz *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],GCD@@#==1&]],{n,0,30}] (* Gus Wiseman, Oct 14 2020 *)

Moebius transform of A000217(n-2).

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = (1 - 3*x + 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 26 2017

Edited by Alois P. Heinz, Feb 08 2011

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.

Original entry on oeis.org

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

Offset: 0

Jack W Grahl, Feb 02 2009, Feb 11 2009

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
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

			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)

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).

For compositions into 4 summands see A156039; also see A156041 and A156042.
Cf. A184957, A071619 (bisection).
A001399(n-2)*2 is the strict case.
A001840(n-2) is the version with opposite relations.
A001840(n-1) is the version with strict opposite relations.
A069905 is the case with strict relations.
A014311 ranks 3-part compositions, with strict case A337453.
A014612 ranks 3-part partitions, with strict case A007304.
Cf. A000217, A001523, A211540, A218004, A220377, A337483, A337484.

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

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*)

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

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
a(n) = round((n+1)*(n+3)/6). - Bill McEachen, Feb 16 2021
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
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

More terms from Alois P. Heinz, Jun 14 2009

A007980 Expansion of (1+x^2)/((1-x)^2*(1-x^3)).

Original entry on oeis.org

1, 2, 4, 7, 10, 14, 19, 24, 30, 37, 44, 52, 61, 70, 80, 91, 102, 114, 127, 140, 154, 169, 184, 200, 217, 234, 252, 271, 290, 310, 331, 352, 374, 397, 420, 444, 469, 494, 520, 547, 574, 602, 631, 660, 690, 721, 752, 784, 817, 850, 884, 919, 954, 990, 1027, 1064

Offset: 0

N. J. A. Sloane

Molien series for ternary self-dual codes over GF(3) of length 12n containing 11...1.
(1+x)*(1+x^2) / ((1-x)*(1-x^2)*(1-x^3)) is the Poincaré series [or Poincare series] (or Molien series) for H^*(O_3(q); F_2).
a(n) is the position of the n-th triangular number in the running sum of the (pseudo-Orloj) sequence 1,2,1,2,1,2,1...., cf. A028355. - Wouter Meeussen, Mar 10 2002
a(n) = [a(n-1) + (number of even terms so far in the sequence)]. Example: 14 is [10 + 4 even terms so far in the sequence (they are 0,2,4,10)]. See A096777 for the same construction with odd integers. - Eric Angelini, Aug 05 2007
The number of partitions of 2*n into at most 3 parts. - Colin Barker, Mar 31 2015
Also a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of a trigonal Rotational Energy Surface. An optimal basis for the expansion follows either decomposition: g1(x) = (1+x)(1+x^2)g2(x) or g1(x) = (1+x^2)x^(-1)g3(x), where g1(x), g2(x), g3(x) are the generating functions for sequences A007980, A001399, A001840. - Bradley Klee, Aug 06 2015
Also a(n) equals the number of linearly-independent terms at 4n-th order in the power series expansion of the symmetrized weight enumerator of a self-dual code of length n over Z4 that contains a vector (+/-)1^n and has all norms divisible by 8. An optimal basis for the expansion follows the decomposition: g1(x) = (1+x)(1+x^2)g2(x) where g1(x), g2(x) are the generating functions for sequences A007980, A001399. (Cf. Calderbank and Sloane, Corollary 5.) - Bradley Klee, Aug 06 2015
Also, a(n) is equal to the number of partitions of 2n+3 of length 3. Letting n=4, there are a(4)=10 partitions of 2n+3=11 of length 3: (9,1,1), (8,2,1), (7,3,1), (7,2,2), (6,4,1), (6,3,2), (5,5,1), (5,4,2), (5,3,3), (4,4,3). - John M. Campbell, Jan 30 2016
a(n) is the number of partitions of n into parts 1 (of two kinds), part 2 (occurring at most once), and parts 3. - Joerg Arndt, Oct 12 2020
Conjecture: a(n) is the maximum number of pieces a triangle can be cut into by n cevians. - Anton Zakharov, Apr 04 2017
Also, a(n) is the number of graphs which are double-triangle descendants of K_5 with n+6 triangles and 3 more vertices than triangles. See Laradji/Mishna/Yeats reference, proposition 3.6 for details. - Karen A. Yeats, Feb 21 2020

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 10*x^4 + 14*x^5 + 19*x^6 + 24*x^7 + ...

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 233.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Raghavendra N. Bhat, Cristian Cobeli, and Alexandru Zaharescu, A lozenge triangulation of the plane with integers, arXiv:2403.10500 [math.NT], 2024.
A. R. Calderbank and N. J. A. Sloane, Double circulant codes over Z_4, J. Algeb. Combin., 6 (1997) 119-131 (Abstract, pdf, ps).
Mohamed Laradji, Marni Mishna, and Karen Yeats, Some results on double triangle descendants of K_5, arXiv:1904.06923 [math.CO], 2019.
C. L. Mallows and N. J. A. Sloane, Weight enumerators of self-orthogonal codes over GF(3), SIAM J. Alg. Discrete Methods, 2 (1981), 452-460.
Paul Tabatabai and Dieter P. Gruber, Knights and Liars on Graphs, J. Int. Seq., Vol. 24 (2021), Article 21.5.8.
Anton Zakharov, cevians.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).
Index entries for Molien series.
Index entries for two-way infinite sequences.

Cf. A000326, A001399, A001840, A002378, A003215, A004396, A005449, A007980.

Cf. A008796, A028355, A049450, A069905, A096777, A192736, A211540, A238705.

with (combinat):seq(count(Partition((2*n+1)), size=3), n=1..56); # Zerinvary Lajos, Mar 28 2008

Table[Ceiling[n (n+1)/3], {n, 56}]
CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^3)),{x,0,60}],x] (* Vincenzo Librandi, Feb 25 2012 *)
a[ n_] := Quotient[ n^2, 3] + n + 1; (* Michael Somos, Aug 23 2015 *)
LinearRecurrence[{2,-1,1,-2,1},{1,2,4,7,10},60] (* Harvey P. Dale, Aug 24 2016 *)

{a(n) = if( n<-1, a(-3-n), polcoeff( (1 + x^2) / ( (1 - x)^2 * (1 - x^3)) + x*O(x^n), n))}; /* Michael Somos, Jun 07 2003 */

{a(n) = n^2\3 + n+1}; /* Michael Somos, Aug 23 2015 */

a(n) = #partitions(2*n, ,[1,3]); \\ Michel Marcus, Feb 12 2016

a(n) = #partitions(2*n+3, ,[3,3]); \\ Michel Marcus, Feb 12 2016

G.f.: (1 + x^2) / ((1 - x)^2 * (1 - x^3)). - Michael Somos, Jun 07 2003
a(n) = a(n-1) + a(n-3) -a(n-4) + 2 = a(-3-n) for all n in Z. - Michael Somos, Jun 07 2003
a(n) = ceiling((n+1)*(n+2)/3). - Paul Boddington, Jan 26 2004
a(n) = A192736(n+1) / (n+1). - Reinhard Zumkeller, Jul 08 2011
From Bruno Berselli, Oct 22 2010: (Start)
a(n) = ((n+1)*(n+2)+(2*cos(2*Pi*n/3)+1)/3)/3 = Sum_{i=1..n+1} A004396(i).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.
a(n) = A002378(n+1)/3 if 3 divides A002378(n+1), a(n) = (A002378(n)+1)/3 otherwise. (End)
a(n) = A001840(n+1) + A001840(n-1). - R. J. Mathar, Aug 23 2015
From Michael Somos, Aug 23 2015: (Start)
Euler transform of length 4 sequence [2, 1, 1, -1].
a(n) = A001399(2*n) = A008796(2*n) = A008796(2*n + 3) = A069905(2*n + 3) = A211540(2*n + 5).
a(2*n)     = A238705(n+1).
a(3*n - 1) = A049451(n).
a(3*n)     = A003215(n).
a(3*n + 1) = A049450(n+1).
2*a(3*n - 1)  = A005449(n).
2*a(3*n + 1)  = A000326(n+1).
a(n+1) - a(n) = A004396(n+2). (End)
a(n) = floor((n^2+3*n+3)/3). - Giacomo Guglieri, May 01 2019
a(n) = A000212(n) + n+1. - Yuchun Ji, Oct 12 2020
Sum_{n>=0} 1/a(n) = (tanh(Pi/(2*sqrt(3)))-1)*Pi/sqrt(3) + 3. - Amiram Eldar, May 20 2023

A060016 Triangle T(n,k) = number of partitions of n into k distinct parts, 1 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

Also number of partitions of n-k(k+1)/2 into at most k parts (not necessarily distinct).

A025147(n) = Sum_{k=2..floor((n+2)/2)} a(n-k+1, k-1). - Reinhard Zumkeller, Nov 04 2007

			Triangle starts
[ 1]  1,
[ 2]  1, 0,
[ 3]  1, 1, 0,
[ 4]  1, 1, 0, 0,
[ 5]  1, 2, 0, 0, 0,
[ 6]  1, 2, 1, 0, 0, 0,
[ 7]  1, 3, 1, 0, 0, 0, 0,
[ 8]  1, 3, 2, 0, 0, 0, 0, 0,
[ 9]  1, 4, 3, 0, 0, 0, 0, 0, 0,
[10]  1, 4, 4, 1, 0, 0, 0, 0, 0, 0,
[11]  1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0,
[12]  1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0,
[13]  1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0,
[14]  1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, ...
T(8,3)=2 since 8 can be written in 2 ways as the sum of 3 distinct positive integers: 5+2+1 and 4+3+1.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.

Alois P. Heinz, Rows n = 1..141, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Columns (offset) include A057427, A004526, A001399, A001400, A001401, etc. Cf. A000009 (row sums), A008289 (without zeros), A030699 (row maximum), A008284 (partition triangle including duplications).

See A008289 for another version.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
T:= proc(n) local l; l:= subsop(1=NULL, b(n, n));
      l[], 0$(n-nops(l))
    end:
seq(T(n), n=1..20);  # Alois P. Heinz, Dec 12 2012

Flatten[Table[Length[Select[IntegerPartitions[n,{k}],Max[Transpose[ Tally[#]][[2]]]==1&]],{n,20},{k,n}]] (* Harvey P. Dale, Feb 27 2012 *)
T[, 1] = 1; T[n, k_] /; 1, ] = 0; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 26 2015 *)

N=16;  q='q+O('q^N);
gf=sum(n=0,N, z^n * q^((n^2+n)/2) / prod(k=1,n, 1-q^k ) );
/* print triangle: */
gf -= 1; /* remove row zero */
P=Pol(gf,'q);
{ for (n=1,N-1,
    p = Pol(polcoeff(P, n),'z);
    p += 'z^(n+1);  /* preserve trailing zeros */
    v = Vec(polrecip(p));
    v = vector(n,k,v[k]); /* trim to size n */
    print(v);
); }
/* Joerg Arndt, Oct 20 2012 */

T(n, k) = T(n-k, k) + T(n-k, k-1) [with T(n, 0)=1 if n=0 and 0 otherwise].

G.f.: Sum_{n>=0} z^n * q^((n^2+n)/2) / Product_{k=1..n} (1-q^k), rows by powers of q, columns by powers of z; includes row 0 (drop term for n=0 for this triangle, see PARI code); setting z=1 gives g.f. for A000009; cf. to g.f. for A072574. - Joerg Arndt, Oct 20 2012

More terms, recurrence, etc. from Henry Bottomley, Mar 26 2001

A253186 Number of connected unlabeled loopless multigraphs with 3 vertices and n edges.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 32, 36, 39, 43, 47, 51, 55, 60, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 119, 126, 132, 139, 146, 153, 160, 168, 175, 183, 191, 199, 207, 216, 224, 233, 242, 251, 260, 270, 279, 289, 299, 309, 319, 330

Offset: 0

Danny Rorabaugh, Mar 23 2015

a(n) is also the number of ways to partition n into 2 or 3 parts.
a(n) is also the dimension of linear space of three-dimensional 2n-homogeneous polynomial vector fields, which have an octahedral symmetry (for a given representation), which are solenoidal, and which are vector fields on spheres. - Giedrius Alkauskas, Sep 30 2017
Apparently a(n) = A244239(n-6) for n > 4. - Georg Fischer, Oct 09 2018
a(n) is also the number of loopless connected n-regular multigraphs with 4 nodes. - Natan Arie Consigli, Aug 09 2019
a(n) is also the number of inequivalent linear [n, k=2] binary codes without 0 columns (see A034253 for more details). - Petros Hadjicostas, Oct 02 2019
Differs from A160138 only by the offset. - R. J. Mathar, May 15 2023
From Allan Bickle, Jul 13 2025: (Start)
a(n) is the number of theta graphs with n-2 vertices, or n-1 edges.  Equivalently, the number of 2-connected graphs with n-2 vertices and n-1 edges.
A theta graph has three paths with length at least 1 identified at their endpoints.  There can at most one path with length 1.
For instance the theta graphs with 6 vertices have paths with lengths (1,2,4), (1,3,3), or (2,2,2), so a(6-2) = 3. (End)

			On vertex set {a, b, c}, every connected multigraph with n = 5 edges is isomorphic to a multigraph with one of the following a(5) = 4 edge multisets: {ab, ab, ab, ab, ac}, {ab, ab, ab, ac, ac}, {ab, ab, ab, ac, bc}, and {ab, ab, ac, ac, bc}.

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG], 2017; see Section 5.3.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k = 2.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,2,2}.]
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; see Eq. (23).
Gordon Royle, Small Multigraphs.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A001399, A003082, A014395, A014396, A014397, A014398, A050535, A076864.
Column k = 3 of A191646 and column k = 2 of A034253.
First differences of A034198 (excepting the first term).
Cf. A213654, A213655, A213668 (theta graphs).

[Floor(n/2) + Floor((n^2 + 6)/12): n in [0..70]]; // Vincenzo Librandi, Mar 24 2015

CoefficientList[Series[- x^2 (x^3 - x - 1) / ((1 - x) (1 - x^2) (1 - x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Mar 24 2015 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 1, 2, 3, 4}, 61] (* Robert G. Wilson v, Oct 11 2017 *)
a[n_]:=Floor[n/2] + Floor[(n^2 + 6)/12]; Array[a, 70, 0] (* Stefano Spezia, Oct 09 2018 *)

[floor(n/2) + floor((n^2 + 6)/12) for n in range(70)]

a(n) = A004526(n) + A069905(n).
a(n) = floor(n/2) + floor((n^2 + 6)/12).
G.f.: x^2*(x^3 - x - 1)/((x - 1)^2*(x^2 - 1)*(x^2 + x + 1)).

A032765 a(n) = floor(n(n+1)(n+2) / (n + n+1 + n+2)) = floor(n*(n+2)/3).

Original entry on oeis.org

0, 1, 2, 5, 8, 11, 16, 21, 26, 33, 40, 47, 56, 65, 74, 85, 96, 107, 120, 133, 146, 161, 176, 191, 208, 225, 242, 261, 280, 299, 320, 341, 362, 385, 408, 431, 456, 481, 506, 533, 560, 587, 616, 645, 674, 705, 736, 767, 800, 833, 866, 901, 936, 971, 1008

Offset: 0

Patrick De Geest, May 15 1998

Satisfies a(n+1) - 2*a(n) + a(n-1) = (2/3)*(1 + w^(n+1) + w^(2*n+2)), a(0)=0 & a(1)=1 where w is the imaginary cubic root of unity. - Robert G. Wilson v, Jun 24 2002
First differences have this pattern: (+1) +1 +1 +3 +3 +3 +5 +5 +5 +7 +7 +7 +9 +9 +9. - Alexandre Wajnberg, Dec 19 2005
In Duistermaat (2010) section 11.3 The Planar Four-Bar Link on page 516: "It follows from (4.3.2) that the number of k-periodic fibers of the QRT automorphism, counted with multiplicities, is equal to nu((tau^S)^k) = 3*n^2 - 1 when k = 3*n, 3*n^2 + 2*n when k = 3*n + 1, 3*n^2 + 4*n + 1 when k = 3*n + 2, for every integer n." - Michael Somos, Mar 14 2023
a(n-1) is the maximum number of edges in a graph with vertices labeled from 1 to n, such that there is no triangle whose sum of vertex labels is divisible by n. - Hoang Xuan Thanh, Jun 11 2025

			G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 11*x^5 + 16*x^6 + 21*x^7 + 26*x^8 + ... - _Michael Somos_, Mar 14 2023
Let n = 5: a(5-1) = 8. Consider the graph G(5) with vertex set {1, 2, 3, 4, 5} and the edge set: E = {12, 23, 34, 45, 51, 13, 24, 35}, which contains 8 edges. The triangles in this graph are: {123}, {234}, {345}, {135}. Their vertex sums are: 1+2+3 = 6, 2+3+4 = 9, 3+4+5 = 12, 1+3+5=9. Since none of these sums are divisible by 5, the graph satisfies the required condition. If we add the edge 14, a new triangle {145} would appear, whose vertex sum is 1+4+5 = 10, divisible by 5. Similarly, if we add the edge 25, the triangle {235} would form, with vertex sum 2+3+5 = 10, also divisible by 5. - _Hoang Xuan Thanh_, Jun 11 2025

Johannes J. Duistermaat, Discrete Integrable Systems, Springer Science+Business Media, 2010.

Johannes Bader, Kobon Triangles.
Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007.
Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007. [Cached copy, with permission]
Taylor Short, The saturation number of carbon nanocones and nanotubes, arXiv:1807.11355 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Kobon Triangle.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A001082, A032766, A000217, A001399.

A032765:=n->floor(n*(n+2)/3); seq(A032765(n), n=0..100); # Wesley Ivan Hurt, Dec 20 2013

Table[Floor[n (n + 1)(n + 2)/(n + (n + 1) + (n + 2))], {n, 0, 55}]
Table[Floor[n (n + 2)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 20 2013 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 2, 5, 8}, 60] (* Harvey P. Dale, Jun 06 2016 *)
Table[(3 n (n + 2) + 2 Cos[2 n Pi/3] + 2 Sqrt[3] Sin[2 n Pi/3] - 2)/9, {n, 0, 20}] (* Eric W. Weisstein, Jul 27 2025 *)
CoefficientList[Series[x (-1 - 2 x^2 + x^3)/((-1 + x)^3 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Jul 27 2025 *)

a(n)=n*(n+2)\3 \\ Charles R Greathouse IV, Jun 11 2015

From Ralf Stephan, May 05 2004: (Start)
a(n) = n^2 - ceiling(n*(n-1)/3).
G.f.: x*(1+2x^2-x^3)/((1+x+x^2)(1-x)^3). (End)
a(n) = floor(n*(n+2)/3). - Saburo Tamura, sent by Alexandre Wajnberg, Dec 19 2005
a(3*n - 1) = 3*n^2 - 1, a(3*n) = 3*n^2 + 2*n, a(3*n + 1) = 3*n^2 + 4*n + 1. - Michael Somos, Mar 14 2023
a(n+5) = A000217(n+5) - A001399(n+3) - A001399(n). - Hoang Xuan Thanh, Jun 11 2025
Sum_{n>=1} (-1)^(n+1)/a(n) = -1/4 + (Pi/(2*sqrt(3)))*(2-cosec(Pi/sqrt(3))). - Amiram Eldar, Jun 18 2025

Name change suggested by Wesley Ivan Hurt, Dec 20 2013

A224697 Number A(n,k) of different ways to divide an n X k rectangle into subsquares, considering only the list of parts; square array A(n,k), n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 4, 5, 7, 5, 4, 1, 1, 1, 1, 4, 7, 9, 9, 7, 4, 1, 1, 1, 1, 5, 8, 14, 11, 14, 8, 5, 1, 1, 1, 1, 5, 10, 17, 20, 20, 17, 10, 5, 1, 1

Offset: 0

Alois P. Heinz, Apr 15 2013

			A(4,5) = 9 because there are 9 ways to divide a 4 X 5 rectangle into subsquares, considering only the list of parts: [20(1 X 1)], [16(1 X 1), 1(2 X 2)], [12(1 X 1), 2(2 X 2)], [11(1 X 1), 1(3 X 3)], [8(1 X 1), 3(2 X 2)], [7(1 X 1), 1(2 X 2), 1(3 X 3)], [4(1 X 1), 4(2 X 2)], [4(1 X 1), 1(4 X 4)], [3(1 X 1), 2(2 X 2), 1(3 X 3)].  There is no way to divide this rectangle into [2(1 X 1), 2(3 X 3)].
Square array A(n,k) begins:
  1, 1, 1,  1,  1,  1,  1,   1,   1,   1, ...
  1, 1, 1,  1,  1,  1,  1,   1,   1,   1, ...
  1, 1, 2,  2,  3,  3,  4,   4,   5,   5, ...
  1, 1, 2,  3,  4,  5,  7,   8,  10,  12, ...
  1, 1, 3,  4,  7,  9, 14,  17,  24,  29, ...
  1, 1, 3,  5,  9, 11, 20,  26,  36,  48, ...
  1, 1, 4,  7, 14, 20, 31,  47,  71,  95, ...
  1, 1, 4,  8, 17, 26, 47,  57, 102, 143, ...
  1, 1, 5, 10, 24, 36, 71, 102, 148, 238, ...
  1, 1, 5, 12, 29, 48, 95, 143, 238, 312, ...

Alois P. Heinz and Christopher Hunt Gribble, Antidiagonals n = 0..27, flattened (first 25 antidiagonals from Alois P. Heinz)

Columns (or rows) k=0+1, 2-5 give: A000012, A008619, A001399, A008763(n+4), A187753.
Main diagonal gives: A034295.
Cf. A225622.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then {} elif n=0 or l=[] then {[]}
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:={};
         for i from k to nops(l) while l[i]=0 do s:=s union
             map(x->sort([x[], 1+i-k]), b(n, [l[j]$j=1..k-1,
                 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, nops(b(n, [0$k])), nops(b(k, [0$n]))):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, l_] := b[n, l] = Module[{i, k, m, s, t}, Which[Max[l] > n, {}, n == 0 || l == {}, {{}}, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[Function[x, Sort[Append[x, 1+i-k]]], b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]] ] ]]]; s]]; a[n_, k_] := If[n >= k, Length @ b[n, Array[0&, k]], Length @ b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)

cp's OEIS Frontend

A055684 Number of different n-pointed stars.

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A337563 Number of pairwise coprime unordered triples of positive integers > 1 summing to n.

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A337604 Number of ordered triples of positive integers summing to n, any two of which have a common divisor > 1.

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A000741 Number of compositions of n into 3 ordered relatively prime parts.

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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.

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A007980 Expansion of (1+x^2)/((1-x)^2*(1-x^3)).

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A060016 Triangle T(n,k) = number of partitions of n into k distinct parts, 1 <= k <= n.

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A032765 a(n) = floor(n(n+1)(n+2) / (n + n+1 + n+2)) = floor(n*(n+2)/3).