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A069905 Number of partitions of n into 3 positive parts.

%I A069905 #143 Mar 13 2025 11:59:02
%S A069905 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,
%T A069905 56,61,65,70,75,80,85,91,96,102,108,114,120,127,133,140,147,154,161,
%U A069905 169,176,184,192,200,208,217,225,234,243,252,261,271,280,290,300,310,320,331,341
%N A069905 Number of partitions of n into 3 positive parts.
%C A069905 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
%C A069905 Also number of partitions of n-3 into parts 1, 2, and 3. - _Joerg Arndt_, Sep 05 2013
%C A069905 Number of incongruent triangles with integer sides that have perimeter 2n-3 (see the Jordan et al. link). - _Freddy Barrera_, Aug 18 2018
%C A069905 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
%C A069905 Number of incongruent triangles formed from any 3 vertices of a regular n-gon. - _Frank M Jackson_, Sep 11 2022
%C A069905 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
%D A069905 Ross Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
%D A069905 Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410.
%D A069905 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.
%H A069905 Washington Bomfim, <a href="/A069905/b069905.txt">Table of n, a(n) for n = 0..10000</a>
%H A069905 Roland Bacher and P. De La Harpe, <a href="https://hal.archives-ouvertes.fr/hal-01285685">Conjugacy growth series of some infinitely generated groups</a>, hal-01285685v2, 2016.
%H A069905 Nick Fischer and Christian Ikenmeyer, <a href="https://arxiv.org/abs/2002.00788">The Computational Complexity of Plethysm Coefficients</a>, arXiv:2002.00788 [cs.CC], 2020.
%H A069905 Ross Honsberger, <a href="/A005044/a005044_1.pdf">Mathematical Gems III</a>, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]
%H A069905 J. H. Jordan, R. Walch and R. J. Wisner, <a href="http://www.jstor.org/stable/2321300">Triangles with integer sides</a>, Amer. Math. Monthly, 86 (1979), 686-689.
%H A069905 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A069905 G.f.: x^3/((1-x)*(1-x^2)*(1-x^3)) = x^3/((1-x)^3*(1+x+x^2)*(1+x)).
%F A069905 a(n) = round(n^2/12).
%F A069905 a(n) = floor((n^2+6)/12). - _Washington Bomfim_, Jul 03 2012
%F A069905 a(-n) = a(n). - _Michael Somos_, Sep 04 2013
%F A069905 a(n) = a(n-1) + A008615(n-1) for n > 0. - _Reinhard Zumkeller_, Apr 28 2014
%F A069905 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
%F A069905 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
%F A069905 a(n) = A008284(n,3). - _Robert A. Russell_, May 13 2018
%F A069905 a(n) = A005044(2*n) = A005044(2*n - 3). - _Freddy Barrera_, Aug 18 2018
%F A069905 a(n) = floor((n^2+k)/12) for all integers k such that 3 <= k <= 7. - _Giacomo Guglieri_, Apr 03 2019
%F A069905 From _Wesley Ivan Hurt_, Apr 19 2019: (Start)
%F A069905 a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} 1.
%F A069905 a(n) = Sum_{i=1..floor(n/3)} floor((n-i)/2) - i + 1. (End)
%F A069905 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
%F A069905 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
%F A069905 From _Ridouane Oudra_, Dec 12 2024: (Start)
%F A069905 a(n) = (n^2 + 2*gcd(n,3) - 3*gcd(n,2))/12.
%F A069905 a(n) = (A198442(n) + A079978(n))/3.
%F A069905 a(n) = A000212(n) - A002620(n).
%F A069905 a(n) = A008133(n+1) - A307018(n+1). (End)
%F A069905 a(n) = (A309511(n) + A309513(n))/3. - _Ray Chandler_, Mar 13 2025
%e A069905 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 + ...
%p A069905 A069905 := n->round(n^2/12): seq(A069905(n), n=0..70);
%t A069905 a[ n_]:= Round[ n^2 / 12] (* _Michael Somos_, Sep 04 2013 *)
%t A069905 CoefficientList[Series[x^3/((1-x)(1-x^2)(1-x^3)), {x, 0, 70}], x] (* _Vincenzo Librandi_, Oct 14 2015 *)
%t A069905 Drop[LinearRecurrence[{1,1,0,-1,-1,1}, Append[Table[0,{5}],1],70],2] (* _Robert A. Russell_, May 17 2018 *)
%o A069905 (PARI) a(n) = floor((n^2+6)/12);  \\ _Washington Bomfim_, Jul 03 2012
%o A069905 (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
%o A069905 (Haskell)
%o A069905 a069905 n = a069905_list !! n
%o A069905 a069905_list = scanl (+) 0 a008615_list
%o A069905 -- _Reinhard Zumkeller_, Apr 28 2014
%o A069905 (Magma) [(n^2+6) div 12: n in [0..70]]; // _Vincenzo Librandi_, Oct 14 2015
%o A069905 (GAP) List([0..70],n->NrPartitions(n,3)); # _Muniru A Asiru_, May 17 2018
%o A069905 (SageMath) [round(n^2/12) for n in range(70)] # _G. C. Greubel_, Apr 03 2019
%Y A069905 Another version of A001399, which is the main entry for this sequence.
%Y A069905 Cf. A005044, A008284, A008615, A026810 (4 positive parts).
%Y A069905 Cf. A198442, A079978, A000212, A002620, A008133, A307018.
%K A069905 nonn,easy
%O A069905 0,6
%A A069905 _N. J. A. Sloane_, May 04 2002