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A322457 Irregular triangle: Row n contains numbers k that have recursively symmetrical partitions having Durfee square with side length n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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.

FORMULA

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

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

EXAMPLE

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.

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

CROSSREFS

Cf.: A190899, A190900, A321223, A322156.

Sequence in context: A176865 A047296 A301759 * A137951 A082694 A004793

Adjacent sequences:  A322454 A322455 A322456 * A322458 A322459 A322460

KEYWORD

nonn,tabf,easy

AUTHOR

Michael De Vlieger, Dec 11 2018

STATUS

approved

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Last modified June 16 13:05 EDT 2021. Contains 345057 sequences. (Running on oeis4.)