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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.

%I A194551 #28 May 16 2020 01:32:07
%S A194551 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 A194551 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 A194551 8,7,6,12,7,6,11,5,10,9,8,15
%N 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.
%C A194551 It appears that if this is written as a triangle (see example) and n >= 3 then row n has the following property:
%C A194551 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 A194551 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 A194551 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 A194551 For more information see A135010.
%H A194551 Robert Price, <a href="/A194551/b194551.txt">Table of n, a(n) for n = 1..56954, 50 rows.</a>
%e A194551 Written as a triangle begins:
%e A194551 1;
%e A194551 2;
%e A194551 3;
%e A194551 4;
%e A194551 5;
%e A194551 3,6;
%e A194551 4,7;
%e A194551 5,4,8;
%e A194551 3,6,5,9;
%e A194551 4,7,6,5,10;
%e A194551 5,4,8,7,6,11;
%e A194551 3,6,5,9,4,8,7,6,12;
%e A194551 4,7,6,5,10,5,9,8,7,13;
%e A194551 5,4,8,7,6,11,6,5,10,9,8,7,14;
%e A194551 ...
%e A194551 Row n has length A008483(n), if n >= 3.
%t A194551 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 A194551 Cf. A008483, A135010, A138121, A141285, A182730, A182731, A182732, A182733, A194550.
%K A194551 nonn
%O A194551 1,2
%A A194551 _Omar E. Pol_, Nov 22 2011