

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 multigraphs with 3 nodes and n edges.
(Formerly M0518 N0186)


90



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
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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:
......16..15..14
....17..5...4...13
..18..6...0...3...12
19..7...1...2...11..26
..20..8...9...10..25
....21..22..23..24
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(n3) 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 1to1 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/(1x^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(n3), n >= 3, is the number of nonequivalent necklaces of 3 beads each of them painted by one of n colors.
The sequence {a(n3), n >= 3} solves the socalled Reis problem about convex kgons in case k=3 (see our comment to A032279).
a(n3) (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 3tuples (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(n3) is the number of the distinct triangles in an ngon, see the Ngaokrajang links.  Kival Ngaokrajang, Mar 16 2013
Also, a(n) is the total number of 5curves coins patterns (5C4S type: 5curves 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


REFERENCES

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III, Problem 33.
N. Benyahia Tani, Z. Yahi, S. Bouroubi, Ordered and nonordered nonisometric convex quadrilaterals inscribed in a regular ngon, Bulletin du Laboratoire Liforce, 01 (2014) 1  9; http://www.liforce.usthb.dz/IMG/pdf/bulletin2014.pdf
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263, #18, P_n^{3}.
S. J. Cyvin et al., Polygonal Systems Including the Corannulene and Coronene Homologs: Novel Applications of Pólya's Theorem, Z. Naturforsch., 52a (1997), 867873.
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.
H. Gupta, Enumeration of incongruent cyclic kgons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964999.
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.
Karl Hermann Struve, Fresnel's Interferenzerscheinungen: Theoretisch und Experimentell Bearbeitet, Dorpat, 1881 (Thesis). [Gives the Round(n^2/12) formula.]
J. H. van Lint, Combinatorial Seminar Eindhoven, Lecture Notes Math., 382 (1974), see pp. 3334.
G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, Sect. 1, Problem 25.
V. Shevelev, Necklaces and convex kgons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629638.
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).
James Tanton, "Young students approach integer triangles", FOCUS 22 no. 5 (2002), 4  6.
W. C. Yang, Maximal and minimal polyhexes, manuscript, 2002.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000
Hamid Afshar, Branislav Cvetkovic, Sabine Ertl, Daniel Grumiller and Niklas Johansson, Conformal ChernSimons holographylock, stock and barrel, arXiv preprint arXiv:1110.5644, 2011
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
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), 686689.
Gerzson Keri and Patric R. J. Ostergard, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.
M. B. Nathanson, Partitions with parts in a finite set, arXiv:math/0002098 [math.NT], 2000.
Kival Ngaokrajang, Distinct triangles in ngon for n = 3..9, Distinct triangles in 45gon
Kival Ngaokrajang, Illustration of 5curves coins patterns
Andrew N. Norris, Higher derivatives and the inverse derivative of a tensorvalued function of a tensor, arXiv:0707.0115, 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, 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)
James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
Eric Weisstein's World of Mathematics, Tripod
Index to sequences with linear recurrences with constant coefficients, signature (1, 1, 0, 1, 1, 1).


FORMULA

G.f.: 1/((1x)*(1x^2)*(1x^3)).
a(n) = round( (n+3)^2/12 ). Note that this cannot be of the form (2i+1)/2, so ties never arise.
a(n) = A008284(n+3, 3), n >= 0.
a(n) = 1 + a(n2) + a(n3)  a(n5) 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(n1)+A008615(n+2) = a(n2) + A008620(n) = a(n3)+A008619(n) = A001840(n+1)  a(n1) = A002620(n+2) A001840(n) = A000601(n)  A000601(n1).  Henry Bottomley, Apr 17 2001
P(n, 3) = 1/72(6*n^279*pcr{1, 1}(2, n)+8*pcr{2, 1, 1}(3, n)) (see Comtet).
Let m>0 and 3<=p<=2 be defined by n = 6*m+p3 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
a(n) = 17/72 + (n+1)*(n+5)/12 + (1)^n/8 + (2/9)*cos(2*n*Pi/3).  Benoit Cloitre, Feb 09 2003
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)=ceil(1/12*n^2+1/2*n)+(n mod 6==0?1:0).  Jon Perry, Jun 17 2003
a(n) = sum(i=0, floor(n/3), 1+floor((n3*i)/2)).  Jon Perry, Jun 27 2003
a(n) = sum_{k=0..n} floor((k+2)/2)*(cos(2*Pi*(nk)/3+Pi/3)/3+sqrt(3)sin(2*Pi*(nk)/3+Pi/3)/3+1/3).  Paul Barry, Apr 16 2005
(m choose 3)_q=(q^m1)*(q^(m1)1)*(q^(m2)1)/((q^31)*(q^21)*(q1)).
a(n) = sum_{k=0..floor(n/2)} floor((3+n2k)/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(n1)+a(n2)a(n4)a(n5)+a(n6).  David Neil McGrath, Feb 14 2015


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(n3)=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


MAPLE

[ seq(1+floor((n^2+6*n)/12), n=0..60) ];
A001399 := 1/(z+1)/(z**2+z+1)/(z1)**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 JeanFrançois 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 *)


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(n1)+Self(n2)Self(n4)Self(n5)+Self(n6): n in [1..80]]; // Vincenzo Librandi, Feb 14 2015


CROSSREFS

Cf. A008724, A003082, A117485, A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496, A008619, A001400, A001401, A128012, A069905, A008615.
Sequence in context: A034092 A211540 * A069905 A008761 A008760 A008759
Adjacent sequences: A001396 A001397 A001398 * A001400 A001401 A001402


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Robert G. Wilson v, Dec 11 2000


STATUS

approved



