A211540 - OEIS

%I #78 Sep 08 2022 08:46:02

%S 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,

%T 52,56,61,65,70,75,80,85,91,96,102,108,114,120,127,133,140,147,154,

%U 161,169,176,184,192,200,208,217,225,234,243,252,261,271,280,290

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

%C For a guide to related sequences, see A211422.

%C Also the number of partitions of n+1 into three parts, where each part > 1. - _Peter Woodward_, May 25 2015

%C 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

%C 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

%C After the first three terms, partial sums of A008615. - _Robert Israel_, Dec 31 2015

%C 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

%H Robert Israel, <a href="/A211540/b211540.txt">Table of n, a(n) for n = 0..10000</a>

%H Clark Kimberling, <a href="https://www.emis.de/journals/JIS/VOL22/Kimberling/kimb9.html">A Combinatorial Classification of Triangle Centers on the Line at Infinity</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.

%H Ece Uslu and Esin Becenen, <a href="https://web.archive.org/web/20160412082710/http://matematikprojesi.com/dosyalar/4360aEce%20Uslu%20Esin%20Becenen%20Tubitak%20Project.pdf">Identical Object Distributions</a>.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).

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

%F a(n) = A069905(n-2) = A001399(n-5) for n >= 5. - _Alois P. Heinz_, Nov 03 2012

%F 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

%F a(n) = A004526(n-2) + a(n-2) for n > 2. - _Ece Uslu_, Esin Becenen, Dec 31 2015

%F G.f.: x^5/(1 - x - x^2 + x^4 + x^5 - x^6). - _Robert Israel_, Dec 31 2015

%F a(n) = Sum_{k=1..floor(n/3)} floor((n-k)/2)-k. - _Wesley Ivan Hurt_, Apr 27 2019

%F From _Gus Wiseman_, Oct 11 2020: (Start)

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

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

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

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

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

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

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

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

%F (End)

%e a(5) = a(6) = 1 with only one ordered triple (5,2,1). - _Michael Somos_, Feb 02 2015

%e 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

%e 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

%e G.f.: x^5 + x^6 + 2*x^7 + 3*x^8 + 4*x^9 + 5*x^10 + 7*x^11 + 8*x^12 + ...

%e From _Gus Wiseman_, Oct 11 2020: (Start)

%e 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.

%e   222  322  332  333  433  443  444  544  554  555  655

%e             422  432  442  533  543  553  644  654  664

%e                  522  532  542  552  643  653  663  754

%e                       622  632  633  652  662  744  763

%e                            722  642  733  743  753  772

%e                                 732  742  752  762  844

%e                                 822  832  833  843  853

%e                                      922  842  852  862

%e                                           932  933  943

%e                                           A22  942  952

%e                                                A32  A33

%e                                                B22  A42

%e                                                     B32

%e                                                     C22

%e 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.

%e   432  532  542  543  643  653  654  754  764  765  865

%e             632  642  652  743  753  763  854  864  874

%e                  732  742  752  762  853  863  873  964

%e                       832  842  843  862  872  954  973

%e                            932  852  943  953  963  982

%e                                 942  952  962  972  A54

%e                                 A32  A42  A43  A53  A63

%e                                      B32  A52  A62  A72

%e                                           B42  B43  B53

%e                                           C32  B52  B62

%e                                                C42  C43

%e                                                D32  C52

%e                                                     D42

%e                                                     E32

%e 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.

%e   321  421  431  432  532  542  543  643  653  654  754

%e             521  531  541  632  642  652  743  753  763

%e                  621  631  641  651  742  752  762  853

%e                       721  731  732  751  761  843  862

%e                            821  741  832  842  852  871

%e                                 831  841  851  861  943

%e                                 921  931  932  942  952

%e                                      A21  941  951  961

%e                                           A31  A32  A42

%e                                           B21  A41  A51

%e                                                B31  B32

%e                                                C21  B41

%e                                                     C31

%e                                                     D21

%e (End)

%p 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):

%p seq(f(i),i=0..100); # _Robert Israel_, Dec 31 2015

%t t[n_] := t[n] = Flatten[Table[-2 w + 3 x + 4 y, {w, n}, {x, n}, {y, n}]]

%t c[n_] := Count[t[n], 0]

%t t = Table[c[n], {n, 0, 80}]  (* A211540 *)

%t FindLinearRecurrence[t]

%t LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 0, 0, 0, 1}, 70] (* _Vincenzo Librandi_, Dec 31 2015 *)

%t Table[Length[Select[IntegerPartitions[n+1,{3}],UnsameQ@@#&]],{n,0,30}] (* _Gus Wiseman_, Oct 05 2020 *)

%o (PARI) {a(n) = round( (n-2)^2 / 12 )}; / * _Michael Somos_, Feb 02 2015 */

%o (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

%o (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

%Y Cf. A001399, A069905, A211422.

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

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

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

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

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

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

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

%K nonn,easy

%O 0,8

%A _Clark Kimberling_, Apr 15 2012