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A001399
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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)
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49
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also number of tripods (trees with exactly 3 leaves) on n vertices - Eric W. Weisstein (eric(AT)weisstein.com), Mar 5, 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 (perry(AT)globalnet.co.uk), Jan 07 2004
Also necklaces 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 (se16(AT)btinternet.com), 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(n-3) is the number of partitions of n into 3 distinct parts, where 0 is allowed as a part. E.g. 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 (perry(AT)globalnet.co.uk), 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 (perry(AT)globalnet.co.uk), 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 (g_m(AT)mcraefamily.com), Feb 07 2005
Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
A117220(n) = a(A003586(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 04 2006
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
a(n) = A026820(n,3) for n>2. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 21 2010]
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)
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REFERENCES
| Hamid Afshar, Branislav Cvetkovic, Sabine Ertl, Daniel Grumiller and Niklas Johansson, Conformal Chern-Simons holography-lock, stock and barrel, Arxiv preprint arXiv:1110.5644, 2011
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}.
S. J. Cyvin et al., Polygonal systems including the corannulene ... homologs ..., Z. Naturforsch., 52a (1997), 867-873.
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 k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
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. Jordan, R. Walch and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly, 86 (1979), 686-689.
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.
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.
V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
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
M. B. Nathanson, Partitions with parts in a finite set
Andrew N. Norris, Higher derivatives and the inverse derivative of a tensor-valued function of a tensor, arXiv:0707.0115, Equation 3.28, p. 10
Jon Perry, More Partition Function
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
V. Shevelev,Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma)(Cf. Section 5)
J. Tanton, Young students approach integer triangles
Eric Weisstein's World of Mathematics, Tripod
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| G.f.: 1/((1-x)*(1-x^2)*(1-x^3)).
a(n) = nearest integer to (n+3)^2/12. Note that this cannot be of the form (2i+1)/2, so ties never arise.
a(n)=1+a(n-2)+a(n-3)-a(n-5). - Michael Somos
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 (se16(AT)btinternet.com), 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).
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 (fvlamoen(AT)hotmail.com), Jul 23 2001
a(n)=17/72+(n+1)*(n+5)/12+(-1)^n/8+(2/9)*cos(2*n*Pi/3) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (perry(AT)globalnet.co.uk), Jun 17 2003
a(n)=sum(i=0, floor(n/3), 1+ceil((n-3*i-1)/2)) - Jon Perry (perry(AT)globalnet.co.uk), Jun 27 2003
a(n)=sum{k=0..floor(n/3), floor((n-3k+2)/2)}; 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 (pbarry(AT)wit.ie), 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-2k)/3)} - Paul Barry (pbarry(AT)wit.ie), Nov 11 2003
a(-6-n)=a(n). - Michael Somos Sep 04 2006
a(n)= 3 * sum_{i=2...n+1} floor(i/2)-floor(i/3) - Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 11 2007
After initial 1 appears identical to integer part of ((n+4)^2 + 4)/12, which is given Norris as 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 + 1. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 03 2007
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EXAMPLE
| Recall that in a necklace the adjacent beads have distinct colors. Let 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. In case 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 nodulo 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]
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MAPLE
| [ seq(1+floor((n^2+6*n)/12), n=0..60) ];
A001399:=-1/(z+1)/(z**2+z+1)/(z-1)**3; [S. 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 (thomas.wieder(AT)t-online.de), Feb 11 2007
with(combstruct):ZL4:=[S, {S=Set(Cycle(Z, card<4))}, unlabeled]:seq(count(ZL4, size=n), n=0..61); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 24 2007
B:=[S, {S = Set(Sequence(Z, 1 <= card), card <=3)}, unlabelled]: seq(combstruct[count](B, size=n), n=0..61); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2009]
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MATHEMATICA
| CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)), {x, 0, 65} ], x ]
Table[ Length[ Select[ Partitions[n], First[ # ] == 3 & ]], {n, 1, 60} ]
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 (russell(AT)post.harvard.edu), Sep 27 2004
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PROG
| (PARI) {a(n)=round((n+3)^2/12)} /* Michael Somos Sep 04 2006 */
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CROSSREFS
| a(6n) = A003215(n), a(6n+1) = A000567(n+1), a(6n+2) = A049450(n+1), a(6n+3) = A033428(n+1), a(6n+4) = A049451(n+1), a(6n+5) = A045944(n+1)
a(n)=A008284(n+3, 3), n >= 0.
Cf. A008724, A003082, A117485. Bisection of A005044.
Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496.
Cf. A008619, A001400, A001401.
Cf. A128012.
Sequence in context: A034163 A034092 * A069905 A008761 A008760 A008759
Adjacent sequences: A001396 A001397 A001398 * A001400 A001401 A001402
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 11 2000
Struve reference from Harrie Grondijs, May 08, 2006
Replaced arXiv URL by a non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 07 2009
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