A211540 Number of ordered triples (w,x,y) with all terms in {1..n} and 2w = 3x + 4y.

0, 0, 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

Offset: 0

Author

Clark Kimberling, Apr 15 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211422.

Also the number of partitions of n+1 into three parts, where each part > 1. - Peter Woodward, May 25 2015

a(n) is also equal to the number of partitions of n+4 into three distinct parts, where each part > 1. - Giovanni Resta, May 26 2015

Number of different distributions of n+1 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Dec 31 2015

After the first three terms, partial sums of A008615. - Robert Israel, Dec 31 2015

For n >= 2, also the number of partitions of n - 2 into 3 parts. The Heinz numbers of these partitions are given by A014612. - Gus Wiseman, Oct 11 2020

Examples

			a(5) = a(6) = 1 with only one ordered triple (5,2,1). - _Michael Somos_, Feb 02 2015
a(11) = 5 Number of different distributions of 11 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
a(1) = a(2) = a(3) = a(4) = a(5) = 0, since with fewer than 6 identical balls there is no such distribution with 3 boxes that holds for 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Dec 31 2015
G.f.: x^5 + x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 8*x^12 + ...
From _Gus Wiseman_, Oct 11 2020: (Start)
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three parts > 1 [Woodward] are the following (A = 10, B = 11, C = 12). The ordered version is A000217(n - 4) and the Heinz numbers are A046316.
  222  322  332  333  433  443  444  544  554  555  655
            422  432  442  533  543  553  644  654  664
                 522  532  542  552  643  653  663  754
                      622  632  633  652  662  744  763
                           722  642  733  743  753  772
                                732  742  752  762  844
                                822  832  833  843  853
                                     922  842  852  862
                                          932  933  943
                                          A22  942  952
                                               A32  A33
                                               B22  A42
                                                    B32
                                                    C22
The a(5) = 1 through a(15) = 14 partitions of n + 4 into three distinct parts > 1 [Resta] are the following (A = 10, B = 11, C = 12, D = 13, E = 14). The ordered version is A211540*6 and the Heinz numbers are A046389.
  432  532  542  543  643  653  654  754  764  765  865
            632  642  652  743  753  763  854  864  874
                 732  742  752  762  853  863  873  964
                      832  842  843  862  872  954  973
                           932  852  943  953  963  982
                                942  952  962  972  A54
                                A32  A42  A43  A53  A63
                                     B32  A52  A62  A72
                                          B42  B43  B53
                                          C32  B52  B62
                                               C42  C43
                                               D32  C52
                                                    D42
                                                    E32
The a(5) = 1 through a(15) = 14 partitions of n + 1 into three distinct parts [Uslu and Becenen] are the following (A = 10, B = 11, C = 12, D = 13). The ordered version is A211540(n)*6 and the Heinz numbers are A007304.
  321  421  431  432  532  542  543  643  653  654  754
            521  531  541  632  642  652  743  753  763
                 621  631  641  651  742  752  762  853
                      721  731  732  751  761  843  862
                           821  741  832  842  852  871
                                831  841  851  861  943
                                921  931  932  942  952
                                     A21  941  951  961
                                          A31  A32  A42
                                          B21  A41  A51
                                               B31  B32
                                               C21  B41
                                                    C31
                                                    D21
(End)
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
• Ece Uslu and Esin Becenen, Identical Object Distributions.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

Cf. A001399, A069905, A211422.

All of the following pertain to 3-part strict partitions.

- A000009 counts these partitions of any length, with non-strict version A000041.

- A007304 gives the Heinz numbers, with non-strict version A014612.

- A101271 counts the relatively prime case, with non-strict version A023023.

- A220377 counts the pairwise coprime case, with non-strict version A307719.

- A337605 counts the pairwise non-coprime case, with non-strict version A337599.

Cf. A000217, A001840, A156040, A284825, A337453, A337483, A337484, A337563.

Programs

Magma

I:=[0,0,0,0,0,1]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015

Maple

f:= gfun:-rectoproc({a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6),seq(a(i)=0,i=0..4),a(5)=1},a(n),remember):
seq(f(i),i=0..100); # Robert Israel, Dec 31 2015

Mathematica

t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 4 y, {w, n}, {x, n}, {y, n}]]
c[n_] := Count[t[n], 0]
t = Table[c[n], {n, 0, 80}]  (* A211540 *)
FindLinearRecurrence[t]
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 1}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[Length[Select[IntegerPartitions[n+1,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Oct 05 2020 *)

PARI

{a(n) = round( (n-2)^2 / 12 )}; / * Michael Somos, Feb 02 2015 */

PARI

concat(vector(5), Vec(x^5/(1-x-x^2+x^4+x^5-x^6) + O(x^100))) \\ Altug Alkan, Jan 10 2016

Formula

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6).

a(n) = A069905(n-2) = A001399(n-5) for n >= 5. - Alois P. Heinz, Nov 03 2012

a(n) = 3*k^2-6*k+3 (for n = 6*k-3), 3*k^2-5*k+2 (for n = 6*k-2), 3*k^2-4*k+1 (for n = 6*k-1), 3*k^2-3*k+1 (for n = 6*k), 3*k^2-2*k (for n = 6*k+1), 3*k^2-k (for n = 6*k+2). - Ece Uslu, Esin Becenen, Dec 31 2015

a(n) = A004526(n-2) + a(n-2) for n > 2. - Ece Uslu, Esin Becenen, Dec 31 2015

G.f.: x^5/(1 - x - x^2 + x^4 + x^5 - x^6). - Robert Israel, Dec 31 2015

a(n) = Sum_{k=1..floor(n/3)} floor((n-k)/2)-k. - Wesley Ivan Hurt, Apr 27 2019

From Gus Wiseman, Oct 11 2020: (Start)

a(n+2) = A069905(n) = A001399(n-3) counts 3-part partitions.

a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part strict partitions.

a(n-1) = A069905(n-3) = A001399(n-6) counts 3-part partitions with no 1's.

a(n-4) = A069905(n-6) = A001399(n-9) counts 3-part strict partitions with no 1's.

A000217(n-2) counts 3-part compositions.

a(n-1)*6 = A069905(n-3)*6 = A001399(n-6)*6 counts 3-part strict compositions.

A000217(n-5) counts 3-part compositions with no 1's.

a(n-4)*6 = A069905(n-6)*6 = A001399(n-9)*6 counts 3-part strict compositions with no 1's.

(End)