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A069905
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Number of partitions of n into 3 positive parts.
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78
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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
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OFFSET
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0,6
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COMMENTS
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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
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
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REFERENCES
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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.
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LINKS
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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.
Ross Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]
Nick Fischer and 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).
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FORMULA
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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
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EXAMPLE
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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 + ...
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MAPLE
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A069905 := n->round(n^2/12): seq(A069905(n), n=0..70);
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MATHEMATICA
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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 *)
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PROG
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(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) 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
(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane, May 04 2002
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STATUS
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approved
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