login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001399 a(n) = 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.
(Formerly M0518 N0186)
132
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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. - Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002

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. Polya/Szego 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-curves coins 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

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. Polya and G. Szego, 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).

W. C. Yang, Maximal and minimal polyhexes, manuscript, 2002.

LINKS

Marius A. Burtea, Table of n, a(n) for n = 0..17501 (terms 0..1000 from T. D. Noe, terms 14001 onwards corrected by Sean A. Irvine, April 25 2019)

Hamid Afshar, Branislav Cvetkovic, Sabine Ertl, Daniel Grumiller, and Niklas Johansson, Conformal Chern-Simons holography-lock, stock and barrel, arXiv preprint arXiv:1110.5644 [hep-th], 2011.

C. Ahmed, P. Martin, and V. Mazorchuk, On the number of principal ideals in d-tonal partition monoids, arXiv preprint arXiv:1503.06718 [math.CO], 2015.

Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68 (2013), 71-79.

N. Benyahia Tani, Z. Yahi, and S. Bouroubi, Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular n-gon, Bulletin du Laboratoire Liforce 01 (2014), 1-9.

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal Systems Including the Corannulene and Coronene Homologs: Novel Applications of Pólya's Theorem, Z. Naturforsch., 52a (1997), 867-873.

Lucia De Luca and Gero Friesecke, Classification of particle numbers with unique Heitmann-Radin minimizer, arXiv:1701.07231 [math-ph], 2017.

H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.

W. G. Harter and C. W. Patterson, Asymptotic eigensolutions of fourth and sixth rank octahedral tensor operators, Journal of Mathematical Physics, 20.7 (1979), 1453-1459. alternate copy

M. D. Hirschhorn and J. A. Sellers, Enumeration of unigraphical partitions, JIS 11 (2008) 08.4.6

R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 352

J. H. Jordan, R. Walch, and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly 86 (1979), 686-689.

Alexander V. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.

Gerzson Keri and Patric R. J. Ostergard, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences 9 (2006), Article 06.4.7.

Mathematics Stack Exchange, What does "pcr" stand for. [This is Comtet's notation for "prime circulator". See pp. 109-110.]

M. B. Nathanson, Partitions with parts in a finite set, arXiv:math/0002098 [math.NT], 2000.

Kival Ngaokrajang, Distinct triangles in n-gon for n = 3..9, Distinct triangles in 45-gon

Kival Ngaokrajang, Illustration of 5-curves coins patterns

Andrew N. Norris, Higher derivatives and the inverse derivative of a tensor-valued function of a tensor, arXiv:0707.0115 [math.SP], 2007; Equation 3.28, p. 10.

Jon Perry, More Partition Function.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Vadim Ponomarenko, Minimal zero sequences of finite cyclic groups, INTEGERS 4 (2004), #A24.

V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math. 35(5) (2004), 629-638.

V. Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).

Karl Hermann Struve, Fresnel's Interferenzerscheinungen: Theoretisch und Experimentell Bearbeitet, Dorpat, 1881 (Thesis). [Gives the Round(n^2/12) formula.]

James Tanton, Young students approach integer triangles, FOCUS 22(5) (2002), 4 - 6.

James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).

Eric Weisstein's World of Mathematics, Tripod.

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Index entries for Molien series

FORMULA

G.f.: 1/((1 - x) * (1 - x^2) * (1 - x^3)).

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. See my comment above. - 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"  (like 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 in and 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. Norris has given, up to a unitary offset of index 'n', floor( (n+3)^2+4 ) )/12, which is the same as floor( (n+3)^2+3 ) )/12 already posted above. - 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

EXAMPLE

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

MAPLE

[ seq(1+floor((n^2+6*n)/12), n=0..60) ];

A001399 := -1/(z+1)/(z**2+z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

for n from 1 to 20 do result:=0: for i from 2 to n+1 do result:=result+(floor(i/2)-floor(i/3)); od; result; od; # Thomas Wieder, Feb 11 2007

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 *)

PROG

(PARI) {a(n) = round((n + 3)^2 / 12)}; /* Michael Somos, Sep 04 2006 */

(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

CROSSREFS

Cf. A008724, A003082, A117485, A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496, A008619, A001400, A001401, A128012, 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.

Sequence in context: A242678 A034092 A211540 * A069905 A008761 A008760

Adjacent sequences:  A001396 A001397 A001398 * A001400 A001401 A001402

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Name edited by Gus Wiseman, Apr 15 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 07:33 EDT 2019. Contains 328252 sequences. (Running on oeis4.)