cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A001399 a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290, 300, 310, 320, 331, 341
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Also number of tripods (trees with exactly 3 leaves) on n vertices. - Eric W. Weisstein, Mar 05 2011
Also number of partitions of n+3 into exactly 3 parts; number of partitions of n in which the greatest part is less than or equal to 3; and the number of nonnegative solutions to b + 2c + 3d = n.
Also a(n) gives number of partitions of n+6 into 3 distinct parts and number of partitions of 2n+9 into 3 distinct and odd parts, e.g., 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3. - Jon Perry, Jan 07 2004
Also bracelets with n+3 beads 3 of which are red (so there are 2 possibilities with 5 beads).
More generally, the number of partitions of n into at most k parts is also the number of partitions of n+k into k positive parts, the number of partitions of n+k in which the greatest part is k, the number of partitions of n in which the greatest part is less than or equal to k, the number of partitions of n+k(k+1)/2 into exactly k distinct positive parts, the number of nonnegative solutions to b + 2c + 3d + ... + kz = n and the number of nonnegative solutions to 2c + 3d + ... + kz <= n. - Henry Bottomley, Apr 17 2001
Also coefficient of q^n in the expansion of (m choose 3)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
From Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) for n > 0 is formed by the folding points (including the initial 1). The spiral begins:
.
85--84--83--82--81--80
/ \
86 56--55--54--53--52 79
/ / \ \
87 57 33--32--31--30 51 78
/ / / \ \ \
88 58 34 16--15--14 29 50 77
/ / / / \ \ \ \
89 59 35 17 5---4 13 28 49 76
/ / / / / \ \ \ \ \
90 60 36 18 6 0 3 12 27 48 75
/ / / / / / / / / / /
91 61 37 19 7 1---2 11 26 47 74
\ \ \ \ / / / /
62 38 20 8---9--10 25 46 73
\ \ \ / / /
63 39 21--22--23--24 45 72
\ \ / /
64 40--41--42--43--44 71
\ /
65--66--67--68--69--70
.
a(p) is maximal number of hexagons in a polyhex with perimeter at most 2p + 6. (End)
a(n-3) is the number of partitions of n into 3 distinct parts, where 0 is allowed as a part. E.g., at n=9, we can write 8+1+0, 7+2+0, 6+3+0, 4+5+0, 1+2+6, 1+3+5 and 2+3+4, which is a(6)=7. - Jon Perry, Jul 08 2003
a(n) gives number of partitions of n+6 into parts <=3 where each part is used at least once (subtract 6=1+2+3 from n). - Jon Perry, Jul 03 2004
This is also the number of partitions of n+3 into exactly 3 parts (there is a 1-to-1 correspondence between the number of partitions of n+3 in which the greatest part is 3 and the number of partitions of n+3 into exactly three parts). - Graeme McRae, Feb 07 2005
Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - Paul Barry, Apr 16 2005
Also, number of triangles that can be created with odd perimeter 3,5,7,9,11,... with all sides whole numbers. Note that triangles with even perimeter can be generated from the odd ones by increasing each side by 1. E.g., a(1) = 1 because perimeter 3 can make {1,1,1} 1 triangle. a(4) = 3 because perimeter 9 can make {1,4,4} {2,3,4} {3,3,3} 3 possible triangles. - Bruce Love (bruce_love(AT)ofs.edu.sg), Nov 20 2006
Also number of nonnegative solutions of the Diophantine equation x+2*y+3*z=n, cf. Pólya/Szegő reference.
From Vladimir Shevelev, Apr 23 2011: (Start)
Also a(n-3), n >= 3, is the number of non-equivalent necklaces of 3 beads each of them painted by one of n colors.
The sequence {a(n-3), n >= 3} solves the so-called Reis problem about convex k-gons in case k=3 (see our comment to A032279).
a(n-3) (n >= 3) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order n with three 1's in every row. (End)
A001399(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w = 2*x+3*y. - Clark Kimberling, Jun 04 2012
Also, for n >= 3, a(n-3) is the number of the distinct triangles in an n-gon, see the Ngaokrajang links. - Kival Ngaokrajang, Mar 16 2013
Also, a(n) is the total number of 5-curve coin patterns (5C4S type: 5 curves covering full 4 coins and symmetry) packing into fountain of coins base (n+3). See illustration in links. - Kival Ngaokrajang, Oct 16 2013
Also a(n) = half the number of minimal zero sequences for Z_n of length 3 [Ponomarenko]. - N. J. A. Sloane, Feb 25 2014
Also, a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of an Octahedral Rotational Energy Surface (cf. Harter & Patterson). - Bradley Klee, Jul 31 2015
Also Molien series for invariants of finite Coxeter groups D_3 and A_3. - N. J. A. Sloane, Jan 10 2016
Number of different distributions of n+6 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Jan 11 2016
a(n) is also the number of partitions of 2*n with <= n parts and no part >= 4. The bijection to partitions of n with no part >= 4 is: 1 <-> 2, 2 <-> 1 + 3, 3 <-> 3 + 3 (observing the order of these rules). The <- direction uses the following fact for partitions of 2*n with <= n parts and no part >=4: for each part 1 there is a part 3, and an even number (including 0) of remaining parts 3. - Wolfdieter Lang, May 21 2019
List of the terms in A000567(n>=1), A049450(n>=1), A033428(n>=1), A049451(n>=1), A045944(n>=1), and A003215(n) in nondecreasing order. List of the numbers A056105(n)-1, A056106(n)-1, A056107(n)-1, A056108(n)-1, A056109(n)-1, and A003215(m) with n >= 1 and m >= 0 in nondecreasing order. Numbers of the forms 3n*(n-1)+1, n*(3n-2), n*(3n-1), 3n^2, n*(3n+1), n*(3n+2) with n >= 1 listed in nondecreasing order. Integers m such that lattice points from 1 through m on a hexagonal spiral starting at 1 forms a convex polygon. - Ya-Ping Lu, Jan 24 2024

Examples

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 10*x^8 + 12*x^9 + ...
Recall that in a necklace the adjacent beads have distinct colors. Suppose we have n colors with labels 1,...,n. Two colorings of the beads are equivalent if the cyclic sequences of the distances modulo n between labels of adjacent colors have the same period. If n=4, all colorings are equivalent. E.g., for the colorings {1,2,3} and {1,2,4} we have the same period {1,1,2} of distances modulo 4. So, a(n-3)=a(1)=1. If n=5, then we have two such periods {1,1,3} and {1,2,2} modulo 5. Thus a(2)=2. - _Vladimir Shevelev_, Apr 23 2011
a(0) = 1, i.e., {1,2,3} Number of different distributions of 6 identical balls to 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Jan 11 2016
a(3) = 3, i.e., {1,2,6}, {1,3,5}, {2,3,4} Number of different distributions of 9 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Jan 11 2016
From _Gus Wiseman_, Apr 15 2019: (Start)
The a(0) = 1 through a(8) = 10 integer partitions of n with at most three parts are the following. The Heinz numbers of these partitions are given by A037144.
  ()  (1)  (2)   (3)    (4)    (5)    (6)    (7)    (8)
           (11)  (21)   (22)   (32)   (33)   (43)   (44)
                 (111)  (31)   (41)   (42)   (52)   (53)
                        (211)  (221)  (51)   (61)   (62)
                               (311)  (222)  (322)  (71)
                                      (321)  (331)  (332)
                                      (411)  (421)  (422)
                                             (511)  (431)
                                                    (521)
                                                    (611)
The a(0) = 1 through a(7) = 8 integer partitions of n + 3 whose greatest part is 3 are the following. The Heinz numbers of these partitions are given by A080193.
  (3)  (31)  (32)   (33)    (322)    (332)     (333)      (3322)
             (311)  (321)   (331)    (3221)    (3222)     (3331)
                    (3111)  (3211)   (3311)    (3321)     (32221)
                            (31111)  (32111)   (32211)    (33211)
                                     (311111)  (33111)    (322111)
                                               (321111)   (331111)
                                               (3111111)  (3211111)
                                                          (31111111)
Non-isomorphic representatives of the a(0) = 1 through a(5) = 5 unlabeled multigraphs with 3 vertices and n edges are the following.
  {}  {12}  {12,12}  {12,12,12}  {12,12,12,12}  {12,12,12,12,12}
            {13,23}  {12,13,23}  {12,13,23,23}  {12,13,13,23,23}
                     {13,23,23}  {13,13,23,23}  {12,13,23,23,23}
                                 {13,23,23,23}  {13,13,23,23,23}
                                                {13,23,23,23,23}
The a(0) = 1 through a(8) = 10 strict integer partitions of n - 6 with three parts are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A007304.
  (321)  (421)  (431)  (432)  (532)  (542)  (543)  (643)   (653)
                (521)  (531)  (541)  (632)  (642)  (652)   (743)
                       (621)  (631)  (641)  (651)  (742)   (752)
                              (721)  (731)  (732)  (751)   (761)
                                     (821)  (741)  (832)   (842)
                                            (831)  (841)   (851)
                                            (921)  (931)   (932)
                                                   (A21)   (941)
                                                           (A31)
                                                           (B21)
The a(0) = 1 through a(8) = 10 integer partitions of n + 3 with three parts are the following. The Heinz numbers of these partitions are given by A014612.
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)
                (311)  (321)  (331)  (422)  (432)  (442)  (533)
                       (411)  (421)  (431)  (441)  (532)  (542)
                              (511)  (521)  (522)  (541)  (551)
                                     (611)  (531)  (622)  (632)
                                            (621)  (631)  (641)
                                            (711)  (721)  (722)
                                                   (811)  (731)
                                                          (821)
                                                          (911)
The a(0) = 1 through a(8) = 10 integer partitions of n whose greatest part is <= 3 are the following. The Heinz numbers of these partitions are given by A051037.
  ()  (1)  (2)   (3)    (22)    (32)     (33)      (322)      (332)
           (11)  (21)   (31)    (221)    (222)     (331)      (2222)
                 (111)  (211)   (311)    (321)     (2221)     (3221)
                        (1111)  (2111)   (2211)    (3211)     (3311)
                                (11111)  (3111)    (22111)    (22211)
                                         (21111)   (31111)    (32111)
                                         (111111)  (211111)   (221111)
                                                   (1111111)  (311111)
                                                              (2111111)
                                                              (11111111)
The a(0) = 1 through a(6) = 7 strict integer partitions of 2n+9 with 3 parts, all of which are odd, are the following. The Heinz numbers of these partitions are given by A307534.
  (5,3,1)  (7,3,1)  (7,5,1)  (7,5,3)   (9,5,3)   (9,7,3)   (9,7,5)
                    (9,3,1)  (9,5,1)   (9,7,1)   (11,5,3)  (11,7,3)
                             (11,3,1)  (11,5,1)  (11,7,1)  (11,9,1)
                                       (13,3,1)  (13,5,1)  (13,5,3)
                                                 (15,3,1)  (13,7,1)
                                                           (15,5,1)
                                                           (17,3,1)
The a(0) = 1 through a(8) = 10 strict integer partitions of n + 3 with 3 parts where 0 is allowed as a part (A = 10):
  (210)  (310)  (320)  (420)  (430)  (530)  (540)  (640)  (650)
                (410)  (510)  (520)  (620)  (630)  (730)  (740)
                       (321)  (610)  (710)  (720)  (820)  (830)
                              (421)  (431)  (810)  (910)  (920)
                                     (521)  (432)  (532)  (A10)
                                            (531)  (541)  (542)
                                            (621)  (631)  (632)
                                                   (721)  (641)
                                                          (731)
                                                          (821)
The a(0) = 1 through a(7) = 7 integer partitions of n + 6 whose distinct parts are 1, 2, and 3 are the following. The Heinz numbers of these partitions are given by A143207.
  (321)  (3211)  (3221)   (3321)    (32221)    (33221)     (33321)
                 (32111)  (32211)   (33211)    (322211)    (322221)
                          (321111)  (322111)   (332111)    (332211)
                                    (3211111)  (3221111)   (3222111)
                                               (32111111)  (3321111)
                                                           (32211111)
                                                           (321111111)
(End)
Partitions of 2*n with <= n parts and no part >= 4: a(3) = 3 from (2^3), (1,2,3), (3^2) mapping to (1^3), (1,2), (3), the partitions of 3 with no part >= 4, respectively. - _Wolfdieter Lang_, May 21 2019

References

  • R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III, Problem 33.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263, #18, P_n^{3}.
  • J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
  • R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
  • J. H. van Lint, Combinatorial Seminar Eindhoven, Lecture Notes Math., 382 (1974), see pp. 33-34.
  • G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, Sect. 1, Problem 25.
  • 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).

Links

Crossrefs

Cf. A008724, A003082, A117485, A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496, A008619, A001400, A001401, A069905, A008615, row 3 of A192517.
Molien series for finite Coxeter groups D_3 through D_12 are A001399, A051263, A266744-A266751.
Cf. A007304, A014612, A037144, A051037, A080193, A143207, A307534.

Programs

  • Haskell
    a001399 = p [1,2,3] where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 28 2013
  • Magma
    I:=[1,1,2,3,4,5]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..80]]; // Vincenzo Librandi, Feb 14 2015
  • Magma
    [#RestrictedPartitions(n,{1,2,3}): n in [0..62]]; // Marius A. Burtea, Jan 06 2019
  • Magma
    [Round((n+3)^2/12): n in [0..70]]; // Marius A. Burtea, Jan 06 2019
  • Maple
    A001399 := proc(n)
    round( (n+3)^2/12) ;
end proc:
seq(A001399(n),n=0..40) ;
with(combstruct):ZL4:=[S,{S=Set(Cycle(Z,card<4))}, unlabeled]:seq(count(ZL4,size=n),n=0..61); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=3)},unlabelled]: seq(combstruct[count](B, size=n), n=0..61); # Zerinvary Lajos, Mar 21 2009
  • Mathematica
    CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)), {x, 0, 65} ], x ]
Table[ Length[ IntegerPartitions[n, 3]], {n, 0, 61} ] (* corrected by Jean-François Alcover, Aug 08 2012 *)
k = 3; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,4,5},70] (* Harvey P. Dale, Jun 21 2012 *)
a[ n_] := With[{m = Abs[n + 3] - 3}, Length[ IntegerPartitions[ m, 3]]]; (* Michael Somos, Dec 25 2014 *)
k=3 (* Number of red beads in bracelet problem *);CoefficientList[Series[(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)
Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Apr 15 2019 *)
  • PARI
    {a(n) = round((n + 3)^2 / 12)}; /* Michael Somos, Sep 04 2006 */
  • Python
    [round((n+3)**2 / 12) for n in range(0,62)] # Ya-Ping Lu, Jan 24 2024

Formula

G.f.: 1/((1 - x) * (1 - x^2) * (1 - x^3)) = -1/((x+1)*(x^2+x+1)*(x-1)^3); Simon Plouffe in his 1992 dissertation
a(n) = round((n + 3)^2/12). Note that this cannot be of the form (2*i + 1)/2, so ties never arise.
a(n) = A008284(n+3, 3), n >= 0.
a(n) = 1 + a(n-2) + a(n-3) - a(n-5) for all n in Z. - Michael Somos, Sep 04 2006
a(n) = a(-6 - n) for all n in Z. - Michael Somos, Sep 04 2006
a(6*n) = A003215(n), a(6*n + 1) = A000567(n + 1), a(6*n + 2) = A049450(n + 1), a(6*n + 3) = A033428(n + 1), a(6*n + 4) = A049451(n + 1), a(6*n + 5) = A045944(n + 1).
a(n) = a(n-1) + A008615(n+2) = a(n-2) + A008620(n) = a(n-3) + A008619(n) = A001840(n+1) - a(n-1) = A002620(n+2) - A001840(n) = A000601(n) - A000601(n-1). - Henry Bottomley, Apr 17 2001
P(n, 3) = (1/72) * (6*n^2 - 7 - 9*pcr{1, -1}(2, n) + 8*pcr{2, -1, -1}(3, n)) (see Comtet). [Here "pcr" stands for "prime circulator" and it is defined on p. 109 of Comtet, while the formula appears on p. 110. - Petros Hadjicostas, Oct 03 2019]
Let m > 0 and -3 <= p <= 2 be defined by n = 6*m+p-3; then for n > -3, a(n) = 3*m^2 + p*m, and for n = -3, a(n) = 3*m^2 + p*m + 1. - Floor van Lamoen, Jul 23 2001
72*a(n) = 17 + 6*(n+1)*(n+5) + 9*(-1)^n - 8*A061347(n). - Benoit Cloitre, Feb 09 2003
From Jon Perry, Jun 17 2003: (Start)
a(n) = 6*t(floor(n/6)) + (n%6) * (floor(n/6) + 1) + (n mod 6 == 0?1:0), where t(n) = n*(n+1)/2.
a(n) = ceiling(1/12*n^2 + 1/2*n) + (n mod 6 == 0?1:0).
[Here "n%6" means "n mod 6" while "(n mod 6 == 0?1:0)" means "if n mod 6 == 0 then 1, else 0" (as in C).]
(End)
a(n) = Sum_{i=0..floor(n/3)} 1 + floor((n - 3*i)/2). - Jon Perry, Jun 27 2003
a(n) = Sum_{k=0..n} floor((k + 2)/2) * (cos(2*Pi*(n - k)/3 + Pi/3)/3 + sqrt(3) * sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3). - Paul Barry, Apr 16 2005
(m choose 3)_q = (q^m-1) * (q^(m-1) - 1) * (q^(m-2) - 1)/((q^3 - 1) * (q^2 - 1) * (q - 1)).
a(n) = Sum_{k=0..floor(n/2)} floor((3 + n - 2*k)/3). - Paul Barry, Nov 11 2003
A117220(n) = a(A003586(n)). - Reinhard Zumkeller, Mar 04 2006
a(n) = 3 * Sum_{i=2..n+1} floor(i/2) - floor(i/3). - Thomas Wieder, Feb 11 2007
Identical to the number of points inside or on the boundary of the integer grid of {I, J}, bounded by the three straight lines I = 0, I - J = 0 and I + 2J = n. - Jonathan Vos Post, Jul 03 2007
a(n) = A026820(n,3) for n > 2. - Reinhard Zumkeller, Jan 21 2010
Euler transform of length 3 sequence [ 1, 1, 1]. - Michael Somos, Feb 25 2012
a(n) = A005044(2*n + 3) = A005044(2*n + 6). - Michael Somos, Feb 25 2012
a(n) = A000212(n+3) - A002620(n+3). - Richard R. Forberg, Dec 08 2013
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - David Neil McGrath, Feb 14 2015
a(n) = floor((n^2+3)/12) + floor((n+2)/2). - Giacomo Guglieri, Apr 02 2019
From Devansh Singh, May 28 2020: (Start)
Let p(n, 3) be the number of 3-part integer partitions in which every part is > 0.
Then for n >= 3, p(n, 3) is equal to:
(n^2 - 1)/12 when n is odd and 3 does not divide n.
(n^2 + 3)/12 when n is odd and 3 divides n.
(n^2 - 4)/12 when n is even and 3 does not divide n.
(n^2)/12 when n is even and 3 divides n.
For n >= 3, p(n, 3) = a(n-3). (End)
a(n) = floor(((n+3)^2 + 4)/12). - Vladimír Modrák, Zuzana Soltysova, Dec 08 2020
Sum_{n>=0} 1/a(n) = 15/4 - Pi/(2*sqrt(3)) + Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - Amiram Eldar, Sep 29 2022
E.g.f.: exp(-x)*(9 + exp(2*x)*(47 + 42*x + 6*x^2) + 16*exp(x/2)*cos(sqrt(3)*x/2))/72. - Stefano Spezia, Mar 05 2023
a(6n) = 1+6*A000217(n); Sum_{i=1..n} a(6*i) = A000578(n+1). - David García Herrero, May 05 2024

Extensions

Name edited by Gus Wiseman, Apr 15 2019

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

Original entry on oeis.org

1, 1, 3, 8, 23, 66, 212, 686, 2389, 8682, 33160, 132277, 550835, 2384411, 10709827, 49782637, 238998910, 1182772364, 6023860266, 31525780044, 169316000494, 932078457785, 5253664040426, 30290320077851, 178480713438362, 1073918172017297
Offset: 0

Views

Author

Vladeta Jovovic, Dec 29 1999

Keywords

Comments

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017
Row sums of the triangle defined by the Multiset Transformation of A076864,
1 ;
0 1;
0 2 1;
0 5 2 1;
0 12 8 2 1;
0 33 22 8 2 1;
0 103 72 26 8 2 1;
0 333 229 87 26 8 2 1;
0 1183 782 295 92 26 8 2 1;
0 4442 2760 1036 315 92 26 8 2 1;
0 17576 10270 3735 1129 321 92 26 8 2 1;
0 72810 39770 13976 4117 1154 321 92 26 8 2 1;
0 314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;
- R. J. Mathar, Jul 18 2017
Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

Examples

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123)
  (12)(13)(23)
  (1)(1)(23)(23)
  (1)(2)(3)(123)
  (1)(2)(13)(23)
  (1)(1)(2)(3)(23)
  (1)(1)(2)(2)(3)(3)
(End)

References

  • Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).

Links

Crossrefs

Cf. A001399, A003082, A014395, A014396, A014397, A014398.
Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.
Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.

Programs

Formula

a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - Max Alekseyev, Oct 18 2017
Euler transform of A076864. - Andrew Howroyd, Oct 23 2019

Extensions

More terms from Sean A. Irvine, Oct 02 2011

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

Views

Author

Danny Rorabaugh, Mar 23 2015

Keywords

Comments

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)

Examples

			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}.

Links

Crossrefs

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).

Programs

  • Magma
    [Floor(n/2) + Floor((n^2 + 6)/12): n in [0..70]]; // Vincenzo Librandi, Mar 24 2015
  • Mathematica
    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 *)
  • Sage
    [floor(n/2) + floor((n^2 + 6)/12) for n in range(70)]

Formula

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)).

A192517 Table read by antidiagonals: T(n,k) = number of multigraphs with n vertices and k edges, with no loops allowed (n >= 1, k >= 0).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 3, 6, 4, 1, 0, 1, 1, 3, 7, 11, 5, 1, 0, 1, 1, 3, 8, 17, 18, 7, 1, 0, 1, 1, 3, 8, 21, 35, 32, 8, 1, 0, 1, 1, 3, 8, 22, 52, 76, 48, 10, 1, 0, 1, 1, 3, 8, 23, 60, 132, 149, 75, 12, 1, 0
Offset: 1

Views

Author

Alberto Tacchella, Jul 03 2011

Keywords

Comments

Rows converge to sequence A050535, i.e. T(n,k) = A050535(k) for n >= 2k.

Examples

			Table begins:
[1,0,0,0,0,0,0,0,0,...],
[1,1,1,1,1,1,1,1,1,...],
[1,1,2,3,4,5,7,8,10,...],
[1,1,3,6,11,18,32,48,75,...],
[1,1,3,7,17,35,76,149,291,...],
[1,1,3,8,21,52,132,313,741,...],
[1,1,3,8,22,60,173,471,1303,...],
[1,1,3,8,23,64,197,588,1806,...],
...

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 171.

Links

Crossrefs

Cf. A008406, A191646, A003082 (row 4), A014395 (row 5), A014396 (row 6).

Programs

  • PARI
    \\ See A191646 for G function.
R(n)={Mat(vectorv(n, k, concat([1], G(k, n-1))))}
{ my(A=R(10)); for(n=1, #A, for(k=1, #A, print1(A[n,k], ", "));print) } \\ Andrew Howroyd, May 14 2018

A014395 Number of multigraphs with 5 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 7, 17, 35, 76, 149, 291, 539, 974, 1691, 2874, 4730, 7620, 11986, 18485, 27944, 41550, 60744, 87527, 124338, 174403, 241650, 331153, 448987, 602853, 801943, 1057615, 1383343, 1795578, 2313595, 2960656, 3763879, 4755505, 5972927, 7460196, 9267980
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 650.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
  • J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.

Links

Crossrefs

Row 5 of A192517.
Cf. A001399, A003082, A014396, A014397, A014398.

Programs

  • Mathematica
    CoefficientList[Series[PairGroupIndex[SymmetricGroup[5],s]/.Table[s[i]->1/(1-x^i),{i,1,Binomial[5,2]}],{x,0,30}],x] (* Geoffrey Critzer, Oct 14 2012 *)
  • PARI
    concat([1], G(5, 40)) \\ See A191646 for G. - Andrew Howroyd, Mar 15 2020

Formula

G.f.: (x^21 + x^20 + 5*x^19 + 8*x^18 + 14*x^17 + 22*x^16 + 32*x^15 + 40*x^14 + 39*x^13 + 47*x^12 + 36*x^11 + 36*x^10 + 25*x^9 + 21*x^8 + 12*x^7 + 11*x^6 + 4*x^5 + 4*x^4 + x^3 + x^2 - x + 1)/((x^6 - 1)*(x^5 - 1)^2*(x^4 - 1)^2*(x^3 - 1)^2*(x - 1)^3*(x + 1)).

Extensions

More terms from Vladeta Jovovic, Dec 23 1999

A014396 Number of loopless multigraphs with 6 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 8, 21, 52, 132, 313, 741, 1684, 3711, 7895, 16310, 32604, 63363, 119745, 220546, 396428, 696750, 1198812, 2022503, 3349574, 5452496, 8732932, 13776366, 21423968, 32872642, 49804323, 74560913, 110369469, 161639227
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 650.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
  • J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.

Links

Crossrefs

Row 6 of A192517.
Cf. A001399, A003082, A014395, A014397, A014398.

Programs

  • Mathematica
    CoefficientList[Series[PairGroupIndex[SymmetricGroup[6],s]/.Table[s[i]->1/(1-x^i),{i,1,Binomial[6,2]}],{x,0,30}],x] (* Geoffrey Critzer, Oct 14 2012 *)
  • PARI
    concat([1], G(6, 40)) \\ See A191646 for G. - Andrew Howroyd, Mar 15 2020

Extensions

More terms and better description from Vladeta Jovovic, Dec 29 1999

A014397 Number of loopless multigraphs with 7 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 173, 471, 1303, 3510, 9234, 23574, 58464, 140340, 326792, 738090, 1619321, 3455129, 7180856, 14555856, 28819926, 55808840, 105834657, 196779279, 359124362, 643976482, 1135731758, 1971734302, 3372477533
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 650.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).

Links

Crossrefs

Row 7 of A192517.
Cf. A001399, A003082, A014395, A014396, A014398.

Programs

Extensions

More terms and better description from Vladeta Jovovic, Dec 29 1999

A014398 Number of loopless multigraphs with 8 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 8, 23, 64, 197, 588, 1806, 5509, 16677, 49505, 143761, 406091, 1114890, 2970964, 7685972, 19311709, 47170674, 112123118, 259662333, 586583731, 1294143065, 2791716176, 5895027869, 12198014683, 24758285639, 49339306519
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • CRC Handbook of Combinatorial Designs, 1996, p. 650.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).

Links

Crossrefs

Row 8 of A192517.
Cf. A001399, A003082, A014395, A014396, A014397.

Programs

Extensions

More terms and better description from Vladeta Jovovic, Dec 29 1999

A328652 Number of unlabeled loopless multigraphs with n edges covering four vertices.

Original entry on oeis.org

0, 1, 3, 7, 13, 25, 40, 65, 99, 146, 208, 294, 399, 538, 711, 926, 1188, 1513, 1896, 2361, 2910, 3557, 4312, 5199, 6214, 7392, 8739, 10276, 12019, 14002, 16224, 18732, 21537, 24669, 28152, 32031, 36309, 41047, 46263, 51997, 58282, 65176, 72688, 80894, 89820, 99518
Offset: 1

Views

Author

Andrew Howroyd, Oct 23 2019

Keywords

Links

Crossrefs

Column k=4 of A309936.
Cf. A001399, A003082.

Programs

  • Mathematica
    LinearRecurrence[{2,0,0,-2,-2,3,0,3,-2,-2,0,0,2,-1},{0,1,3,7,13,25,40,65,99,146,208,294,399,538},50] (* Harvey P. Dale, Mar 06 2021 *)
  • PARI
    concat([0], Vec((1 + x + x^2 - x^3 + x^4 - 2*x^5 + 2*x^6)/((1 - x)^6*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2) + O(x^40)))

Formula

a(n) = A003082(n) - A001399(n).
a(n) = 2*a(n-1) - 2*a(n-4) - 2*a(n-5) + 3*a(n-6) + 3*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-13) - a(n-14) for n > 14.
G.f.: x^2*(1 + x + x^2 - x^3 + x^4 - 2*x^5 + 2*x^6)/((1 - x)^6*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2).
Showing 1-9 of 9 results.

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