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).
72, 2450, 648, 1225, 120050, 450, 3969, 581042, 211250, 20808, 9801, 30625
Offset: 1
Examples
For a(1) = 72 SRS(a(1)) is unimodal: 12321. For a(2) = 2450 the center part of SRS(a(2)) is not unimodal: 1212123212121. 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. 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): i\j 1 2 3 4 5 6 7 ... ------------------------------------------------------------- 1 | 72 648 450 20808 27378 11250 1996002 2 | 2450 120050 211250 61250 81225 5281250 1531250 3 | 1225 581042 >10^7 354025 >10^7 148225 442225 4 | 3969 30625 321489 127449 1500625 2393209 5 | 9801 6175225 765625 1375929 648025 6 | 4809249 88209 2082249 983961 7 | 385641 1185921 159201 >10^7 8 | 5461569 3470769 7144929 9 | 7177041 10 | 8497225 ...
Programs
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Mathematica
(* widthPattern[ ] and its support functions are defined in A376829 *) 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] t377668[581042, {4, 4}] (* initial 4x4 section except for T(3, 3) > 10^7 *)
Comments