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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.
2
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.
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 *)
KEYWORD
nonn
AUTHOR
Omar E. Pol, Nov 22 2011
STATUS
approved