0,8

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

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).

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)

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)

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

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 */

(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

(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

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.

Sequence in context: A034163 A242678 A034092 * A001399 A069905 A008761

Adjacent sequences: A211537 A211538 A211539 * A211541 A211542 A211543

nonn,easy

Clark Kimberling, Apr 15 2012

approved