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

%I A322457 #10 Dec 31 2018 02:42:24
%S A322457 1,3,4,6,10,12,9,11,15,17,21,27,16,18,22,24,28,34,36,38,40,48,25,27,
%T A322457 31,33,37,43,45,47,49,55,57,59,61,75,36,38,42,44,48,54,56,58,60,66,68,
%U A322457 70,72,78,80,84,86,90,108,49,51,55,57,61,67,69,71,73,79,81
%N A322457 Irregular triangle: Row n contains numbers k that have recursively symmetrical partitions having Durfee square with side length n.
%C A322457 For all n, n^2 <= k <= 3*n^2.
%C A322457 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.
%H A322457 Michael De Vlieger, <a href="/A322457/b322457.txt">Table of n, a(n) for n = 1..10391</a> (rows 1 <= n <= 36, flattened)
%H A322457 Michael De Vlieger, <a href="/A322457/a322457.png">Illustration for A322457</a>
%H A322457 Michael De Vlieger, <a href="/A322457/a322457_1.png">Annotated plot of k <= 1200 in rows n <= 34</a>, vertical exaggeration 12x.
%F A322457 First term of row n = n^2 = A000290(n).
%F A322457 Last term of row n = 3*n^2 = 3*A000290(n).
%e A322457 Triangle begins:
%e A322457 Row 1:   1,  3;
%e A322457 Row 2:   4,  6, 10, 12;
%e A322457 Row 3:   9, 11, 15, 17, 21, 27;
%e A322457 Row 4:  16, 18, 22, 24, 28, 34, 36, 38, 40, 48;
%e A322457         ...
%e A322457 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}:
%e A322457 (2,2), sum 4, or in terms of squares, {2}:
%e A322457    11
%e A322457    11;
%e A322457 (3,2,1), sum 6, or in terms of squares, {2,1}:
%e A322457    112
%e A322457    11
%e A322457    2;
%e A322457 (4,3,2,1), sum 10, or in terms of squares, {2,1,1}:
%e A322457    1123
%e A322457    113
%e A322457    23
%e A322457    3;
%e A322457 (4,4,2,2), sum 12, or in terms of squares, {2,2}:
%e A322457    1122
%e A322457    1122
%e A322457    22
%e A322457    22.
%t A322457 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
%Y A322457 Cf.: A190899, A190900, A321223, A322156.
%K A322457 nonn,tabf,easy
%O A322457 1,2
%A A322457 _Michael De Vlieger_, Dec 11 2018