A354762 Irregular triangle read by rows in which the row n lists the partition of n into the minimum number of triangular parts.

0, 1, 1, 1, 3, 3, 1, 3, 1, 1, 6, 6, 1, 6, 1, 1, 6, 3, 10, 10, 1, 10, 1, 1, 10, 3, 10, 3, 1, 15, 15, 1, 15, 1, 1, 15, 3, 15, 3, 1, 15, 3, 1, 1, 21, 21, 1, 21, 1, 1, 21, 3, 21, 3, 1, 21, 3, 1, 1, 21, 6, 28, 28, 1, 28, 1, 1, 28, 3, 28, 3, 1, 28, 3, 1, 1, 28, 6, 28, 6, 1

Offset

0,5

Comments

The representation of the partitions (for fixed n) is as (weakly) decreasing list of the parts.

Links

Table of n, a(n) for n=0..80.

Example

The irregular triangle begins:
     0;
     1;
     1, 1;
     3;
     3, 1;
     3, 1, 1;
     6;
     6, 1;
     6, 1, 1;
     6, 3;
    10;
    10, 1;
    10, 1, 1;
    10, 3;
    10, 3, 1;
    15;
    15, 1;
    15, 1, 1;
    15, 3;
    15, 3, 1;
    15, 3, 1, 1;
    ...

Mathematica

nmax=35; With[{triangulars = Table[k(k+1)/2, {k, (Sqrt[1+8*nmax]-1)/2}]}, Flatten[Join[{0}, Flatten[Table[First[IntegerPartitions[n, All, triangulars, 1]], {n, nmax}]]]]] (* improved by Stefano Spezia, May 10 2026 *)

Crossrefs

Cf. A000041, A000217, A007294.

Cf. A001477 (row sums), A057944 (1st column), A057945 (row lengths).

Cf. A354763.

Sequence in context: A215596 A387305 A268676 * A180938 A123561 A323626

Adjacent sequences: A354759 A354760 A354761 * A354763 A354764 A354765

Keyword

nonn,easy,tabf

Author

Stefano Spezia, Jun 06 2022

Status

approved