%I A001399 M0518 N0186
%S A001399 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,
%T A001399 70,75,80,85,91,96,102,108,114,120,127,133,140,147,154,161,169,176,184,
%U A001399 192,200,208,217,225,234,243,252,261,271,280,290,300,310,320,331,341
%N A001399 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.
%C A001399 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 A001399 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
%C A001399 Also necklaces with n+3 beads 3 of which are red (so there are 2 possibilities
with 5 beads).
%C A001399 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
%C A001399 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 A001399 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 A001399 ......16..15..14
%C A001399 ....17..5...4...13
%C A001399 ..18..6...0...3...12
%C A001399 19..7...1...2...11..26
%C A001399 ..20..8...9...10..25
%C A001399 ....21..22..23..24
%C A001399 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
%C A001399 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
%C A001399 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
%C A001399 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
%C A001399 Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - Paul Barry
(pbarry(AT)wit.ie), Apr 16 2005
%C A001399 A117220(n) = a(A003586(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Mar 04 2006
%C A001399 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
%D A001399 R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math.
Soc., 1963; Chapter III, Problem 33.
%D A001399 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263,
#18, P_n^{3}.
%D A001399 S. J. Cyvin et al., Polygonal systems including the corannulene ... homologs
..., Z. Naturforsch., 52a (1997), 867-873.
%D A001399 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press,
2004; p. 517.
%D A001399 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables,
Vol. 4, Cambridge Univ. Press, 1958, p. 2.
%D A001399 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY,
1973, p. 88, (4.1.18).
%D A001399 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers.
3rd ed., Oxford Univ. Press, 1954, p. 275.
%D A001399 J. H. Jordan, R. Walch and R. J. Wisner, Triangles with integer sides,
Amer. Math. Monthly, 86 (1979), 686-689.
%D A001399 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.
%D A001399 J. H. van Lint, Combinatorial Seminar Eindhoven, Lecture Notes Math.,
382 (1974), see pp. 33-34.
%D A001399 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001399 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001399 Karl Hermann Struve, Fresnel's Interferenzerscheinungen: Theoretisch
und Experimentell Bearbeitet, Dorpat, 1881 (Thesis). [Gives the Round(n^2/
12) formula.]
%D A001399 W. C. Yang, Maximal and minimal polyhexes, manuscript, 2002.
%H A001399 T. D. Noe, <a href="b001399.txt">Table of n, a(n) for n=0..1000</a>
%H A001399 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A001399 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=352">
Encyclopedia of Combinatorial Structures 352</a>
%H A001399 M. B. Nathanson, <a href="http://arXiv.org/abs/math.NT/0002098">Partitions
with parts in a finite set</a>
%H A001399 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, Equation 3.28, p. 10
%H A001399 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/
morepartitionfunction.htm">More Partition Function</a>
%H A001399 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A001399 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001399 J. Tanton, <a href="http://www.maa.org/features/integertriangles.pdf">
Young students approach integer triangles</a>
%H A001399 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A001399 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)).
%F A001399 a(n) = nearest integer to (n+3)^2/12. Note that this cannot be of the
form (2i+1)/2, so ties never arise.
%F A001399 a(n)=1+a(n-2)+a(n-3)-a(n-5). - Michael Somos
%F A001399 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
%F A001399 P(n, 3) = 1/72(6*n^2-7-9*pcr{1, -1}(2, n)+8*pcr{2, -1, -1}(3, n)) (see
Comtet).
%F A001399 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
%F A001399 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
%F A001399 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
%F A001399 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
%F A001399 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
%F A001399 (m choose 3)_q=(q^m-1)*(q^(m-1)-1)*(q^(m-2)-1)/((q^3-1)*(q^2-1)*(q-1))
%F A001399 a(n)=sum{k=0..floor(n/2), floor((3+n-2k)/3)} - Paul Barry (pbarry(AT)wit.ie),
Nov 11 2003
%F A001399 a(-6-n)=a(n). - Michael Somos Sep 04 2006
%F A001399 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
%F A001399 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
%e A001399 (3 choose 3)_q = 1, (4 choose 3)_q = q^3 + q^2 + q + 1, (5 choose 3)_q
= q^6 + q^5 + 2*q^4 + 2*q^3 + 2*q^2 + q + 1, (6 choose 3)_q = q^9
+ q^8 + 2*q^7 + 3*q^6 + 3*q^5 + 3*q^4 + 3*q^3 + 2*q^2 + q + 1 so
the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so
on.
%p A001399 [ seq(1+floor((n^2+6*n)/12), n=0..60) ];
%p A001399 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
%p A001399 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
%p A001399 A001399:=-1/(z+1)/(z**2+z+1)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]
%p A001399 with (combinat):seq(count(Partition((3^2+n)), size=3), n=-6..55); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2008
%p A001399 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]
%t A001399 CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)), {x, 0, 65}
], x ]
%t A001399 Table[ Length[ Select[ Partitions[n], First[ # ] == 3 & ]], {n, 1, 60}
]
%t A001399 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
%o A001399 (PARI) {a(n)=round((n+3)^2/12)} /* Michael Somos Sep 04 2006 */
%Y A001399 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)
%Y A001399 a(n)=A008284(n+3, 3), n >= 0.
%Y A001399 Cf. A008724, A003082, A117485. Bisection of A005044.
%Y A001399 Cf. A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228,
A036496.
%Y A001399 Cf. A008619, A001400, A001401.
%Y A001399 Cf. A128012.
%Y A001399 Sequence in context: A034162 A034163 A034092 this_sequence A069905 A008761
A008760
%Y A001399 Adjacent sequences: A001396 A001397 A001398 this_sequence A001400 A001401
A001402
%K A001399 nonn,easy,nice
%O A001399 0,3
%A A001399 N. J. A. Sloane (njas(AT)research.att.com).
%E A001399 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 11 2000
%E A001399 Struve reference from Harrie Grondijs, May 08, 2006
%E A001399 Replaced arXiv URL by a non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Oct 07 2009
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