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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.)