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%I M0518 N0186 #438 Feb 16 2024 14:30:47
%S 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,
%T 65,70,75,80,85,91,96,102,108,114,120,127,133,140,147,154,161,169,176,
%U 184,192,200,208,217,225,234,243,252,261,271,280,290,300,310,320,331,341
%N 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.
%C Also number of tripods (trees with exactly 3 leaves) on n vertices. - _Eric W. Weisstein_, Mar 05 2011
%C 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.
%C 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
%C Also bracelets with n+3 beads 3 of which are red (so there are 2 possibilities with 5 beads).
%C 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
%C 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
%C From Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002: (Start)
%C 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:
%C .
%C 85--84--83--82--81--80
%C / \
%C 86 56--55--54--53--52 79
%C / / \ \
%C 87 57 33--32--31--30 51 78
%C / / / \ \ \
%C 88 58 34 16--15--14 29 50 77
%C / / / / \ \ \ \
%C 89 59 35 17 5---4 13 28 49 76
%C / / / / / \ \ \ \ \
%C 90 60 36 18 6 0 3 12 27 48 75
%C / / / / / / / / / / /
%C 91 61 37 19 7 1---2 11 26 47 74
%C \ \ \ \ / / / /
%C 62 38 20 8---9--10 25 46 73
%C \ \ \ / / /
%C 63 39 21--22--23--24 45 72
%C \ \ / /
%C 64 40--41--42--43--44 71
%C \ /
%C 65--66--67--68--69--70
%C .
%C a(p) is maximal number of hexagons in a polyhex with perimeter at most 2p + 6. (End)
%C 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
%C 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
%C 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
%C Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - _Paul Barry_, Apr 16 2005
%C 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
%C Also number of nonnegative solutions of the Diophantine equation x+2*y+3*z=n, cf. Pólya/Szegő reference.
%C From _Vladimir Shevelev_, Apr 23 2011: (Start)
%C 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.
%C 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).
%C 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)
%C 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
%C 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
%C 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
%C Also a(n) = half the number of minimal zero sequences for Z_n of length 3 [Ponomarenko]. - _N. J. A. Sloane_, Feb 25 2014
%C 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
%C Also Molien series for invariants of finite Coxeter groups D_3 and A_3. - _N. J. A. Sloane_, Jan 10 2016
%C 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
%C 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
%C 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
%D R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III, Problem 33.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263, #18, P_n^{3}.
%D J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
%D R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
%D J. H. van Lint, Combinatorial Seminar Eindhoven, Lecture Notes Math., 382 (1974), see pp. 33-34.
%D G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, Sect. 1, Problem 25.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Marius A. Burtea, <a href="/A001399/b001399.txt">Table of n, a(n) for n = 0..17501</a> (terms 0..1000 from T. D. Noe, terms 14001 onwards corrected by Sean A. Irvine, April 25 2019)
%H Hamid Afshar, Branislav Cvetkovic, Sabine Ertl, Daniel Grumiller, and Niklas Johansson, <a href="http://arxiv.org/abs/1110.5644">Conformal Chern-Simons holography-lock, stock and barrel</a>, arXiv preprint arXiv:1110.5644 [hep-th], 2011.
%H C. Ahmed, P. Martin, and V. Mazorchuk, <a href="http://arxiv.org/abs/1503.06718">On the number of principal ideals in d-tonal partition monoids</a>, arXiv preprint arXiv:1503.06718 [math.CO], 2015.
%H Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, <a href="http://ftp.math.uni-rostock.de/pub/romako/heft68/bouroubi68.html">Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon</a>, Rostocker Math. Kolloq. 68 (2013), 71-79.
%H N. Benyahia Tani, Z. Yahi, and S. Bouroubi, <a href="https://liforce.usthb.dz/sites/default/files/2020-11/article1.pdf">Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular n-gon</a>, Bulletin du Laboratoire Liforce 01 (2014), 1-9.
%H Jonathan Bloom and Nathan McNew, <a href="https://arxiv.org/abs/1908.03953">Counting pattern-avoiding integer partitions</a>, arXiv:1908.03953 [math.CO], 2019.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, <a href="http://zfn.mpdl.mpg.de/data/Reihe_A/52/ZNA-1997-52a-0867.pdf">Polygonal Systems Including the Corannulene and Coronene Homologs: Novel Applications of Pólya's Theorem</a>, Z. Naturforsch., 52a (1997), 867-873.
%H Lucia De Luca and Gero Friesecke, <a href="https://arxiv.org/abs/1701.07231">Classification of particle numbers with unique Heitmann-Radin minimizer</a>, arXiv:1701.07231 [math-ph], 2017.
%H Nick Fischer and Christian Ikenmeyer, <a href="https://arxiv.org/abs/2002.00788">The Computational Complexity of Plethysm Coefficients</a>, arXiv:2002.00788 [cs.CC], 2020.
%H H. Gupta, <a href="http://web.archive.org/web/20200806162943/http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a66_964.pdf">Enumeration of incongruent cyclic k-gons</a>, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
%H W. G. Harter and C. W. Patterson, <a href="http://dx.doi.org/10.1063/1.524199">Asymptotic eigensolutions of fourth and sixth rank octahedral tensor operators</a>, Journal of Mathematical Physics, 20.7 (1979), 1453-1459. <a href="http://www.cwpatterson.com/pdfs/ab-Asymptotic%20Eigensolutions%20of%20Fourth%20and%20Sixth%20Rank%20Octahe.pdf">alternate copy</a>
%H M. D. Hirschhorn and J. A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Sellers/sellers44.html">Enumeration of unigraphical partitions</a>, JIS 11 (2008) 08.4.6.
%H R. Honsberger, <a href="/A005044/a005044_1.pdf">Mathematical Gems III</a>, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=352">Encyclopedia of Combinatorial Structures 352</a>.
%H J. H. Jordan, R. Walch, and R. J. Wisner, <a href="http://www.jstor.org/stable/2321300">Triangles with integer sides</a>, Amer. Math. Monthly 86 (1979), 686-689.
%H Alexander V. Karpov, <a href="https://wp.hse.ru/data/2018/04/04/1164595187/188EC2018.pdf">An Informational Basis for Voting Rules</a>, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.
%H Gerzson Keri and Patric R. J. Östergård, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Keri/keri6.html">The Number of Inequivalent (2R+3,7)R Optimal Covering Codes</a>, Journal of Integer Sequences 9 (2006), Article 06.4.7.
%H Clark Kimberling, <a href="https://www.emis.de/journals/JIS/VOL22/Kimberling/kimb9.html">A Combinatorial Classification of Triangle Centers on the Line at Infinity</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
%H Axel Kleinschmidt and Valentin Verschinin, <a href="https://doi.org/10.1007/JHEP09(2017)155">Tetrahedral modular graph functions</a>, J. High Energy Phys. 2017, No. 9, Paper No. 155, 38 p. (2017), eq (3.40).
%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/1816132/what-does-pcr-stand-for">What does "pcr" stand for</a>. [This is Comtet's notation for "prime circulator". See pp. 109-110.]
%H M. B. Nathanson, <a href="https://arxiv.org/abs/math/0002098">Partitions with parts in a finite set</a>, arXiv:math/0002098 [math.NT], 2000.
%H Kival Ngaokrajang, <a href="/A001399/a001399.jpg">Distinct triangles in n-gon for n = 3..9</a>, <a href="/A001399/a001399_1.jpg">Distinct triangles in 45-gon</a>
%H Kival Ngaokrajang, <a href="/A001399/a001399.pdf">Illustration of 5-curves coins patterns</a>
%H Andrew N. Norris, <a href="http://arXiv.org/abs/0707.0115">Higher derivatives and the inverse derivative of a tensor-valued function of a tensor</a>, arXiv:0707.0115 [math.SP], 2007; Equation 3.28, p. 10.
%H Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/morepartitionfunction.htm">More Partition Function</a>.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Vadim Ponomarenko, <a href="http://www.emis.de/journals/INTEGERS/papers/e24/e24.Abstract.html">Minimal zero sequences of finite cyclic groups</a>, INTEGERS 4 (2004), #A24.
%H Vladimir Shevelev, <a href="http://web.archive.org/web/20200722171019/http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/2000c4e8_629.pdf">Necklaces and convex k-gons</a>, Indian J. Pure and Appl. Math. 35(5) (2004), 629-638.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1104.4051">Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma)</a>, arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).
%H Devansh Singh and S. N. Mishra, <a href="https://www.ripublication.com/gjpam19/gjpamv15n6_07.pdf">Representing K-parts Integer Partitions</a>, Global Journal of Pure and Applied Mathematics (GJPAM), Volume 15 Number 6 (2019), pp. 889-907.
%H Karl Hermann Struve, <a href="https://doi.org/10.1002/andp.18822510106">Fresnel's Interferenzerscheinungen: Theoretisch und Experimentell Bearbeitet</a>, Dorpat, 1881 (Thesis). [Gives the Round(n^2/12) formula.]
%H James Tanton, <a href="http://www.maa.org/sites/default/files/pdf/pubs/mayjun02web.pdf">Young students approach integer triangles</a>, FOCUS 22(5) (2002), 4 - 6.
%H James Tanton, <a href="http://www.jamestanton.com/?p=1413">Integer Triangles</a>, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
%H Richard Vale and Shayne Waldron, <a href="https://doi.org/10.1007/s00041-015-9443-9">The construction of G-invariant finite tight frames</a>, J. Four. Anal. Applic. 22 (2016), 1097-1120.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Tripod.html">Tripod</a>.
%H Gus Wiseman, <a href="/A001399/a001399.txt">Number of ordered triples of distinct positive integers summing to n</a>.
%H Gus Wiseman, <a href="/A001399/a001399_1.txt">Number of non-unimodal triples of distinct positive integers summing to n</a>.
%H Gus Wiseman, <a href="/A001399/a001399_2.txt">Number of unimodal triples of distinct positive integers summing to n</a>.
%H Winston C. Yang, <a href="https://pages.cs.wisc.edu/~ferris/techreports/02-04.pdf">Maximal and minimal polyhexes</a>, 2002.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>.
%F G.f.: 1/((1 - x) * (1 - x^2) * (1 - x^3)).
%F a(n) = round((n + 3)^2/12). Note that this cannot be of the form (2*i + 1)/2, so ties never arise.
%F a(n) = A008284(n+3, 3), n >= 0.
%F a(n) = 1 + a(n-2) + a(n-3) - a(n-5) for all n in Z. - _Michael Somos_, Sep 04 2006
%F a(n) = a(-6 - n) for all n in Z. - _Michael Somos_, Sep 04 2006
%F 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).
%F 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
%F 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]
%F 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
%F 72*a(n) = 17 + 6*(n+1)*(n+5) + 9*(-1)^n - 8*A061347(n). - _Benoit Cloitre_, Feb 09 2003
%F From _Jon Perry_, Jun 17 2003: (Start)
%F 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.
%F a(n) = ceiling(1/12*n^2 + 1/2*n) + (n mod 6 == 0?1:0).
%F [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).]
%F (End)
%F a(n) = Sum_{i=0..floor(n/3)} 1 + floor((n - 3*i)/2). - _Jon Perry_, Jun 27 2003
%F 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
%F (m choose 3)_q = (q^m-1) * (q^(m-1) - 1) * (q^(m-2) - 1)/((q^3 - 1) * (q^2 - 1) * (q - 1)).
%F a(n) = Sum_{k=0..floor(n/2)} floor((3 + n - 2*k)/3). - _Paul Barry_, Nov 11 2003
%F A117220(n) = a(A003586(n)). - _Reinhard Zumkeller_, Mar 04 2006
%F a(n) = 3 * Sum_{i=2..n+1} floor(i/2) - floor(i/3). - _Thomas Wieder_, Feb 11 2007
%F 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
%F a(n) = A026820(n,3) for n > 2. - _Reinhard Zumkeller_, Jan 21 2010
%F Euler transform of length 3 sequence [ 1, 1, 1]. - _Michael Somos_, Feb 25 2012
%F a(n) = A005044(2*n + 3) = A005044(2*n + 6). - _Michael Somos_, Feb 25 2012
%F a(n) = A000212(n+3) - A002620(n+3). - _Richard R. Forberg_, Dec 08 2013
%F a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - _David Neil McGrath_, Feb 14 2015
%F a(n) = floor((n^2+3)/12) + floor((n+2)/2). - _Giacomo Guglieri_, Apr 02 2019
%F From _Devansh Singh_, May 28 2020: (Start)
%F Let p(n, 3) be the number of 3-part integer partitions in which every part is > 0.
%F Then for n >= 3, p(n, 3) is equal to:
%F (n^2 - 1)/12 when n is odd and 3 does not divide n.
%F (n^2 + 3)/12 when n is odd and 3 divides n.
%F (n^2 - 4)/12 when n is even and 3 does not divide n.
%F (n^2)/12 when n is even and 3 divides n.
%F For n >= 3, p(n, 3) = a(n-3). (End)
%F a(n) = floor(((n+3)^2 + 4)/12). - _Vladimír Modrák_, _Zuzana Soltysova_, Dec 08 2020
%F 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
%F 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
%e 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 + ...
%e 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
%e 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
%e 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
%e From _Gus Wiseman_, Apr 15 2019: (Start)
%e 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.
%e () (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (211) (221) (51) (61) (62)
%e (311) (222) (322) (71)
%e (321) (331) (332)
%e (411) (421) (422)
%e (511) (431)
%e (521)
%e (611)
%e 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.
%e (3) (31) (32) (33) (322) (332) (333) (3322)
%e (311) (321) (331) (3221) (3222) (3331)
%e (3111) (3211) (3311) (3321) (32221)
%e (31111) (32111) (32211) (33211)
%e (311111) (33111) (322111)
%e (321111) (331111)
%e (3111111) (3211111)
%e (31111111)
%e Non-isomorphic representatives of the a(0) = 1 through a(5) = 5 unlabeled multigraphs with 3 vertices and n edges are the following.
%e {} {12} {12,12} {12,12,12} {12,12,12,12} {12,12,12,12,12}
%e {13,23} {12,13,23} {12,13,23,23} {12,13,13,23,23}
%e {13,23,23} {13,13,23,23} {12,13,23,23,23}
%e {13,23,23,23} {13,13,23,23,23}
%e {13,23,23,23,23}
%e 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.
%e (321) (421) (431) (432) (532) (542) (543) (643) (653)
%e (521) (531) (541) (632) (642) (652) (743)
%e (621) (631) (641) (651) (742) (752)
%e (721) (731) (732) (751) (761)
%e (821) (741) (832) (842)
%e (831) (841) (851)
%e (921) (931) (932)
%e (A21) (941)
%e (A31)
%e (B21)
%e 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.
%e (111) (211) (221) (222) (322) (332) (333) (433) (443)
%e (311) (321) (331) (422) (432) (442) (533)
%e (411) (421) (431) (441) (532) (542)
%e (511) (521) (522) (541) (551)
%e (611) (531) (622) (632)
%e (621) (631) (641)
%e (711) (721) (722)
%e (811) (731)
%e (821)
%e (911)
%e 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.
%e () (1) (2) (3) (22) (32) (33) (322) (332)
%e (11) (21) (31) (221) (222) (331) (2222)
%e (111) (211) (311) (321) (2221) (3221)
%e (1111) (2111) (2211) (3211) (3311)
%e (11111) (3111) (22111) (22211)
%e (21111) (31111) (32111)
%e (111111) (211111) (221111)
%e (1111111) (311111)
%e (2111111)
%e (11111111)
%e 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.
%e (5,3,1) (7,3,1) (7,5,1) (7,5,3) (9,5,3) (9,7,3) (9,7,5)
%e (9,3,1) (9,5,1) (9,7,1) (11,5,3) (11,7,3)
%e (11,3,1) (11,5,1) (11,7,1) (11,9,1)
%e (13,3,1) (13,5,1) (13,5,3)
%e (15,3,1) (13,7,1)
%e (15,5,1)
%e (17,3,1)
%e 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):
%e (210) (310) (320) (420) (430) (530) (540) (640) (650)
%e (410) (510) (520) (620) (630) (730) (740)
%e (321) (610) (710) (720) (820) (830)
%e (421) (431) (810) (910) (920)
%e (521) (432) (532) (A10)
%e (531) (541) (542)
%e (621) (631) (632)
%e (721) (641)
%e (731)
%e (821)
%e 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.
%e (321) (3211) (3221) (3321) (32221) (33221) (33321)
%e (32111) (32211) (33211) (322211) (322221)
%e (321111) (322111) (332111) (332211)
%e (3211111) (3221111) (3222111)
%e (32111111) (3321111)
%e (32211111)
%e (321111111)
%e (End)
%e 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
%p [ seq(1+floor((n^2+6*n)/12), n=0..60) ];
%p A001399 := -1/(z+1)/(z**2+z+1)/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation
%p 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
%p with(combstruct):ZL4:=[S,{S=Set(Cycle(Z,card<4))}, unlabeled]:seq(count(ZL4,size=n),n=0..61); # _Zerinvary Lajos_, Sep 24 2007
%p 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
%t CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)), {x, 0, 65} ], x ]
%t Table[ Length[ IntegerPartitions[n, 3]], {n, 0, 61} ] (* corrected by _Jean-François Alcover_, Aug 08 2012 *)
%t 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 *)
%t LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,4,5},70] (* _Harvey P. Dale_, Jun 21 2012 *)
%t a[ n_] := With[{m = Abs[n + 3] - 3}, Length[ IntegerPartitions[ m, 3]]]; (* _Michael Somos_, Dec 25 2014 *)
%t 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 *)
%t Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&]],{n,0,30}] (* _Gus Wiseman_, Apr 15 2019 *)
%o (PARI) {a(n) = round((n + 3)^2 / 12)}; /* _Michael Somos_, Sep 04 2006 */
%o (Haskell)
%o a001399 = p [1,2,3] where
%o p _ 0 = 1
%o p [] _ = 0
%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o -- _Reinhard Zumkeller_, Feb 28 2013
%o (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
%o (Magma) [#RestrictedPartitions(n,{1,2,3}): n in [0..62]]; // _Marius A. Burtea_, Jan 06 2019
%o (Magma) [Round((n+3)^2/12): n in [0..70]]; // _Marius A. Burtea_, Jan 06 2019
%o (Python) [print(round((n+3)**2/12), end = ', ') for n in range(0,62)] # _Ya-Ping Lu_, Jan 24 2024
%Y Cf. A008724, A003082, A117485, A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496, A008619, A001400, A001401, A069905, A008615, row 3 of A192517.
%Y Molien series for finite Coxeter groups D_3 through D_12 are A001399, A051263, A266744-A266751.
%Y Cf. A007304, A014612, A037144, A051037, A080193, A143207, A307534.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Name edited by _Gus Wiseman_, Apr 15 2019
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