0,3

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

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.

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.

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.

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).

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

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

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.

J. Tanton, Young students approach integer triangles

Index entries for sequences related to linear recurrences with constant coefficients

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

(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.

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

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

A001399:=-1/(z+1)/(z**2+z+1)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

with (combinat):seq(count(Partition((3^2+n)), size=3), n=-6..55); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 28 2008

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]

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

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

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: A034162 A034163 A034092 this_sequence A069905 A008761 A008760

Adjacent sequences: A001396 A001397 A001398 this_sequence A001400 A001401 A001402

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

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