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%I A377668 #12 Feb 19 2025 12:20:17
%S A377668 72,2450,648,1225,120050,450,3969,581042,211250,20808,9801,30625
%N A377668 Square array read by antidiagonals upwards: T(i,j), i, j >= 1, is the smallest number m such that the symmetric presentation of sigma, SRS(m), has maximum width 3, consists of 2*i-1 parts and has 2*j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).
%C A377668 Maximum width 3 can occur an odd number of times in the width pattern of SRS(m) only for numbers m in this sequence for which SRS(m) has an odd number of parts. In that case width 3 must occur at the diagonal of SRS(m). However, the center part of SRS(m) need not be unimodal.
%e A377668 For a(1) = 72 SRS(a(1)) is unimodal: 12321.
%e A377668 For a(2) = 2450 the center part of SRS(a(2)) is not unimodal: 1212123212121.
%e A377668 For a(11) = 9801 SRS(a(11)) consists of 9 unimodal parts with maximum width in successive parts nondecreasing to the center part of SRS(a(11)); its width pattern is: 1 0 1 0 1 2 1 0 1 2 1 0 1 2 3 2 1 0 1 2 1 0 1 2 1 0 1 0 1.
%e A377668 Ragged upper left hand section of table T(i, j) = m, numbers m <= 10^7, rows i denoting 2*i-1 parts in SRS(m) and columns j denoting 2*j-1 occurrences of width 3 in the width pattern of SRS(m):
%e A377668 i\j 1 2 3 4 5 6 7 ...
%e A377668 -------------------------------------------------------------
%e A377668 1 | 72 648 450 20808 27378 11250 1996002
%e A377668 2 | 2450 120050 211250 61250 81225 5281250 1531250
%e A377668 3 | 1225 581042 >10^7 354025 >10^7 148225 442225
%e A377668 4 | 3969 30625 321489 127449 1500625 2393209
%e A377668 5 | 9801 6175225 765625 1375929 648025
%e A377668 6 | 4809249 88209 2082249 983961
%e A377668 7 | 385641 1185921 159201 >10^7
%e A377668 8 | 5461569 3470769 7144929
%e A377668 9 | 7177041
%e A377668 10 | 8497225
%e A377668 ...
%t A377668 (* widthPattern[ ] and its support functions are defined in A376829 *)
%t A377668 t377668[b_, {r_, c_}] := Module[{t=ConstantArray[0, {r, c}], k, wP, c3, p3}, For[k=1, k<=b, k++, wP=widthPattern[k]; If[Max[wP]==3, c3=Count[wP, 3]; If[OddQ[c3]&&c3+1<=2c, c3=(c3+1)/2; p3=Length[Select[SplitBy[wP, #!=0&], First[#]!=0&]]; If[OddQ[p3]&&p3+1<=2r, p3=(p3+1)/2; If[t[[p3, c3]]==0, t[[p3, c3]]=k]]]]]; t]
%t A377668 t377668[581042, {4, 4}] (* initial 4x4 section except for T(3, 3) > 10^7 *)
%Y A377668 Subsequence of A376829.
%Y A377668 Cf. A237591, A237593, A249223, A341969, A377667.
%K A377668 nonn,tabl,more
%O A377668 1,1
%A A377668 _Hartmut F. W. Hoft_, Nov 03 2024