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%I #26 Dec 10 2016 00:24:01
%S 2,2,1,1,4,1,1,6,1,1,6,1,2,1,3,3,1,1,8,1,1,3,3,1,6,1,10,1,1,10,1,1,10,
%T 1,3,3,1,3,3,1,1,12,1,4,1,3,3,1,3,3,1,14,1,1,14,1,1,14,1,3,3,1,3,3,1,
%U 6,1,16,1,1,3,3,1,3,3,1,16,1,1,16,1,1,7,7,1,14,1,3,3,1,1,18,1,6,1,3,10,3,1,3,3,1,7,7
%N Irregular triangle read by rows in which the n-th row lists the number of legs in the parts of the symmetric representation of sigma(n).
%C The legs are those line segments in the parts of the symmetric representation of sigma(n) that bound a portion of its nonzero area.
%C Blocks of nonzero numbers start and end at odd positions in the rows of triangle A249223 unless a block extends to the end of the row. Therefore, the number of legs in any part of the symmetric representation of sigma(n) is odd when A237271(n) is even, and odd except for the middle part when A237271(n) is odd.
%e Since row 14 of triangle A249223 is 1 1 1 0 the symmetric representation of sigma(14) has two parts of three legs each and row 14 in this triangle is 3 3.
%e Since row 15 of triangle A249223 is 1 0 1 1 2 the symmetric representation of sigma(15) has three parts of 1 leg, 6 legs, and 1 leg, respectively, and row 15 in this triangle is 1 6 1.
%e Irregular triangle of legs of parts:
%e 1: 2
%e 2: 2
%e 3: 1 1
%e 4: 4
%e 5: 1 1
%e 6: 6
%e 7: 1 1
%e 8: 6
%e 9: 1 2 1
%e 10: 3 3
%e 11: 1 1
%e 12: 8
%e 13: 1 1
%e 14: 3 3
%e 15: 1 6 1
%e 16: 10
%e 17: 1 1
%e 18: 10
%e 19: 1 1
%e 20: 10
%e 21: 1 3 3 1
%e ...
%e Illustration of the legs for the symmetric representations of sigma(1)..sigma(24); for comparison see also A237593. The legs of the central parts of the symmetric representation of sigma for 9, 15 and 21 have 3, 3 and 4 parts, and touch the legs of 8, 14 and 20, respectively.
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%e n: 1 2 3 4 5 6 7 8..10..12..14..16..18..20..22..24
%e .
%t (* support functions are defined in A237048 and A262045 *)
%t a279104[n_] := Map[Length, Select[SplitBy[a262045[n], #!=0&], First[#]!=0&]]
%t Flatten[Map[a279104, Range[52]]] (* sequence data for 52 rows *)
%Y Cf. A237271 gives the row lengths.
%Y Cf. A237048, A237270, A237271, A237591, A237593, A245092, A249223, A262045.
%K nonn,tabf
%O 1,1
%A _Hartmut F. W. Hoft_, Dec 06 2016
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