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A069905 Number of partitions of n into 3 positive parts. 60
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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 x<y<z. A one-to-one correspondence between the ordered triples (x,y,z) defined above and the partitions (a,b,c) of n into 3 positive parts is shown by letting x=a-1 and letting z=c+1. - Dennis P. Walsh, Apr 19 2019

REFERENCES

R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410.

D. 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, P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.

R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]

Nick Fischer, Christian Ikenmeyer, The Computational Complexity of Plethysm Coefficients, arXiv:2002.00788 [cs.CC], 2020.

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).

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)

EXAMPLE

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 + ...

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 *)

PROG

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

(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

(PARI) 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

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

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

CROSSREFS

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

Cf. A008615, A026810 (4 positive parts).

Sequence in context: A034092 A211540 A001399 * A008761 A008760 A008759

Adjacent sequences:  A069902 A069903 A069904 * A069906 A069907 A069908

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, May 04 2002

STATUS

approved

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Last modified March 4 20:56 EST 2021. Contains 341807 sequences. (Running on oeis4.)