A194551 - OEIS

%I #28 May 16 2020 01:32:07

%S 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,

%T 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,

%U 8,7,6,12,7,6,11,5,10,9,8,15

%N 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.

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

%C 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).

%C 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).

%C 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).

%C For more information see A135010.

%H Robert Price, <a href="/A194551/b194551.txt">Table of n, a(n) for n = 1..56954, 50 rows.</a>

%e Written as a triangle begins:

%e 1;

%e 2;

%e 3;

%e 4;

%e 5;

%e 3,6;

%e 4,7;

%e 5,4,8;

%e 3,6,5,9;

%e 4,7,6,5,10;

%e 5,4,8,7,6,11;

%e 3,6,5,9,4,8,7,6,12;

%e 4,7,6,5,10,5,9,8,7,13;

%e 5,4,8,7,6,11,6,5,10,9,8,7,14;

%e ...

%e Row n has length A008483(n), if n >= 3.

%t 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 *)

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

%K nonn

%O 1,2

%A _Omar E. Pol_, Nov 22 2011