A069905 Number of partitions of n into 3 positive parts.

0, 0, 0, 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

Offset: 0

Author

N. J. A. Sloane, May 04 2002

Keywords

• nonn
• easy

Comments

Number of binary bracelets of n beads, 3 of them 0. For n >= 3, a(n-3) is the number of binary bracelets of n beads, 3 of them 0, with 00 prohibited. - Washington Bomfim, Aug 27 2008

Also number of partitions of n-3 into parts 1, 2, and 3. - Joerg Arndt, Sep 05 2013

Number of incongruent triangles with integer sides that have perimeter 2n-3 (see the Jordan et al. link). - Freddy Barrera, Aug 18 2018

Number of ordered triples (x,y,z) of nonnegative integers such that x+y+z=n and xDennis P. Walsh, Apr 19 2019

Number of incongruent triangles formed from any 3 vertices of a regular n-gon. - Frank M Jackson, Sep 11 2022

Also a(n-3) for n > 2, otherwise 0 is the number of incongruent scalene triangles formed from the vertices of a regular n-gon. - Frank M Jackson, Nov 27 2022

Examples

			G.f. = x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 + 10*x^11 + ...
		

References

• Ross Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
• Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410.
• Donald E. Knuth, The Art of Computer Programming, vol. 4,fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Links

• Washington Bomfim, Table of n, a(n) for n = 0..10000
• Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
• Nick Fischer and Christian Ikenmeyer, The Computational Complexity of Plethysm Coefficients, arXiv:2002.00788 [cs.CC], 2020.
• Ross Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]
• J. H. Jordan, R. Walch and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly, 86 (1979), 686-689.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

Another version of A001399, which is the main entry for this sequence.

Cf. A005044, A008284, A008615, A026810 (4 positive parts).

Cf. A198442, A079978, A000212, A002620, A008133, A307018.

Programs

GAP

List([0..70],n->NrPartitions(n,3)); # Muniru A Asiru, May 17 2018

Haskell

a069905 n = a069905_list !! n
a069905_list = scanl (+) 0 a008615_list
-- Reinhard Zumkeller, Apr 28 2014

Magma

[(n^2+6) div 12: n in [0..70]]; // Vincenzo Librandi, Oct 14 2015

Maple

A069905 := n->round(n^2/12): seq(A069905(n), n=0..70);

Mathematica

a[ n_]:= Round[ n^2 / 12] (* Michael Somos, Sep 04 2013 *)
CoefficientList[Series[x^3/((1-x)(1-x^2)(1-x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Oct 14 2015 *)
Drop[LinearRecurrence[{1,1,0,-1,-1,1}, Append[Table[0,{5}],1],70],2] (* Robert A. Russell, May 17 2018 *)

PARI

a(n) = floor((n^2+6)/12);  \\ Washington Bomfim, Jul 03 2012

PARI

my(x='x+O('x^70)); concat([0, 0, 0], Vec(x^3/((1-x)*(1-x^2)*(1-x^3)))) \\ Altug Alkan, Oct 14 2015

SageMath

[round(n^2/12) for n in range(70)] # G. C. Greubel, Apr 03 2019

Formula

G.f.: x^3/((1-x)*(1-x^2)*(1-x^3)) = x^3/((1-x)^3*(1+x+x^2)*(1+x)).

a(n) = round(n^2/12).

a(n) = floor((n^2+6)/12). - Washington Bomfim, Jul 03 2012

a(-n) = a(n). - Michael Somos, Sep 04 2013

a(n) = a(n-1) + A008615(n-1) for n > 0. - Reinhard Zumkeller, Apr 28 2014

Let n = 6k + m. Then a(n) = n^2/12 + a(m) - m^2/12. Also, a(n) = 3*k^2 + m*k + a(m). Example: a(35) = a(6*5 + 5) = 35^2/12 + a(5) - 5^2/12 = 102 = 3*5^2 + 5*5 + a(5). - Gregory L. Simay, Oct 13 2015

a(n) = a(n-1) +a(n-2) -a(n-4) -a(n-5) +a(n-6), n>5. - Wesley Ivan Hurt, Oct 16 2015

a(n) = A008284(n,3). - Robert A. Russell, May 13 2018

a(n) = A005044(2*n) = A005044(2*n - 3). - Freddy Barrera, Aug 18 2018

a(n) = floor((n^2+k)/12) for all integers k such that 3 <= k <= 7. - Giacomo Guglieri, Apr 03 2019

From Wesley Ivan Hurt, Apr 19 2019: (Start)

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} 1.

a(n) = Sum_{i=1..floor(n/3)} floor((n-i)/2) - i + 1. (End)

Sum_{n>=3} 1/a(n) = 15/4 + Pi^2/18 - Pi/(2*sqrt(3)) + tanh(Pi/(2*sqrt(3))) * Pi/sqrt(3). - Amiram Eldar, Sep 27 2022

E.g.f.: (8*exp(-x/2)*cos(sqrt(3)*x/2) + (3*x^2 + 3*x - 8)*cosh(x) + (3*x^2 + 3*x + 1)*sinh(x))/36. - Stefano Spezia, Apr 05 2023

From Ridouane Oudra, Dec 12 2024: (Start)

a(n) = (n^2 + 2*gcd(n,3) - 3*gcd(n,2))/12.

a(n) = (A198442(n) + A079978(n))/3.

a(n) = A000212(n) - A002620(n).

a(n) = A008133(n+1) - A307018(n+1). (End)

a(n) = (A309511(n) + A309513(n))/3. - Ray Chandler, Mar 13 2025