A322457 Irregular triangle: Row n contains numbers k that have recursively symmetrical partitions having Durfee square with side length n.

1, 3, 4, 6, 10, 12, 9, 11, 15, 17, 21, 27, 16, 18, 22, 24, 28, 34, 36, 38, 40, 48, 25, 27, 31, 33, 37, 43, 45, 47, 49, 55, 57, 59, 61, 75, 36, 38, 42, 44, 48, 54, 56, 58, 60, 66, 68, 70, 72, 78, 80, 84, 86, 90, 108, 49, 51, 55, 57, 61, 67, 69, 71, 73, 79, 81

Offset: 1

Author

Michael De Vlieger, Dec 11 2018

Keywords

• nonn
• tabf
• easy

Comments

For all n, n^2 <= k <= 3*n^2.

For n > 5, some k may have more than 1 recursively self-conjugate partitions in the same row. For example, k = 90 in row 6 has two recursively self-conjugate partitions (RSCPs) with Durfee square of 6: (12,12,12,9,9,9,6,6,6,3,3,3) and (12,11,11,11,11,7,6,5,5,5,5,1). These RSCPs can be defined by dendritically laying out squares in the series {6,3,3} and {6,5,1} respectively.

Examples

			Triangle begins:
Row 1:   1,  3;
Row 2:   4,  6, 10, 12;
Row 3:   9, 11, 15, 17, 21, 27;
Row 4:  16, 18, 22, 24, 28, 34, 36, 38, 40, 48;
        ...
Row 2 contains the following recursively self-conjugate partitions with Durfee square with side length 2. Below are diagrams that place {2^0, 2^1, 2^2, ... 2^(m-1)} squares of side lengths in S = {k_1, k_2, k_3, ..., k_m}:
(2,2), sum 4, or in terms of squares, {2}:
   11
   11;
(3,2,1), sum 6, or in terms of squares, {2,1}:
   112
   11
   2;
(4,3,2,1), sum 10, or in terms of squares, {2,1,1}:
   1123
   113
   23
   3;
(4,4,2,2), sum 12, or in terms of squares, {2,2}:
   1122
   1122
   22
   22.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10391 (rows 1 <= n <= 36, flattened)
• Michael De Vlieger, Illustration for A322457
• Michael De Vlieger, Annotated plot of k <= 1200 in rows n <= 34, vertical exaggeration 12x.

Crossrefs

Cf.: A190899, A190900, A321223, A322156.

Programs

Mathematica

f[n_] := Block[{w = {n}, c}, c[x_] := Apply[Times, Most@ x - Reverse@ Accumulate@ Reverse@ Rest@ x]; Reap[Do[Which[And[Length@ w == 2, SameQ @@ w], Sow[w]; Break[], Length@ w == 1, Sow[w]; AppendTo[w, 1], c[w] > 0, Sow[w]; AppendTo[w, 1], True, Sow[w]; w = MapAt[1 + # &, Drop[w, -1], -1] ], {i, Infinity}] ][[-1, 1]] ]; Array[Union@ Map[Total@ MapIndexed[#1^2*2^First[#2 - 1] &, #] &, f[#]] &, 7] // Flatten

Formula

First term of row n = n^2 = A000290(n).

Last term of row n = 3*n^2 = 3*A000290(n).