A194551 a(n) is the n-th largest part that are visible in one of the three views of the version "Tree" of the section model of partitions.

1, 2, 3, 4, 5, 3, 6, 4, 7, 5, 4, 8, 3, 6, 5, 9, 4, 7, 6, 5, 10, 5, 4, 8, 7, 6, 11, 3, 6, 5, 9, 4, 8, 7, 6, 12, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 5, 4, 8, 7, 6, 11, 6, 5, 10, 9, 8, 7, 14, 3, 6, 5, 9, 4, 8, 7, 6, 12, 7, 6, 11, 5, 10, 9, 8, 15

Offset

1,2

Comments

It appears that if this is written as a triangle (see example) and n >= 3 then row n has the following property:

If n is congruent to 0 (mod 3) then row n converge to the sequence 3,6,5,9,4,8,7,6,12... in which the records are the numbers >= 3 that are congruent to 0 (mod 3).

If n is congruent to 1 (mod 3) then row n converge to the sequence 4,7,6,5,10,5,9,8,7,13... in which the records are the numbers >= 4 that are congruent to 1 (mod 3).

If n is congruent to 2 (mod 3) then row n converge to the sequence 5,4,8,7,6,11,6,5,10,9,8,7,14... in which the records are the numbers >= 5 that are congruent to 2 (mod 3).

For more information see A135010.

Links

Robert Price, Table of n, a(n) for n = 1..56954, 50 rows.

Example

Written as a triangle begins:
1;
2;
3;
4;
5;
3,6;
4,7;
5,4,8;
3,6,5,9;
4,7,6,5,10;
5,4,8,7,6,11;
3,6,5,9,4,8,7,6,12;
4,7,6,5,10,5,9,8,7,13;
5,4,8,7,6,11,6,5,10,9,8,7,14;
...
Row n has length A008483(n), if n >= 3.

Mathematica

Join[{1}, Table[Drop[l = Last/@DeleteCases[Sort@PadRight[Reverse /@ Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], First@FirstPosition[l, n - 2, {0}]], {n, 2, 15}]] // Flatten  (* Robert Price, May 15 2020 *)

Crossrefs

Cf. A008483, A135010, A138121, A141285, A182730, A182731, A182732, A182733, A194550.

Sequence in context: A215092 A345018 A309079 * A212644 A351285 A341673

Adjacent sequences: A194548 A194549 A194550 * A194552 A194553 A194554

Keyword

nonn

Author

Omar E. Pol, Nov 22 2011

Status

approved