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Revision History for A377668 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

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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).
(history; published version)
#8 by N. J. A. Sloane at Sun Nov 17 07:40:26 EST 2024
STATUS

editing

approved

#7 by N. J. A. Sloane at Sun Nov 17 07:40:23 EST 2024
NAME

Square Array 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).

STATUS

proposed

editing

#6 by Omar E. Pol at Sun Nov 03 17:45:42 EST 2024
STATUS

editing

proposed

#5 by Omar E. Pol at Sun Nov 03 17:44:58 EST 2024
NAME

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 2i2*i-1 parts and has 2j2*j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).

COMMENTS

This sequence is a subsequence of A376829.

EXAMPLE

Ragged upper left hand section of table T(i, j) = m, numbers m <= 10^7, rows i denoting 2i2*i-1 parts in SRS(m) and columns j denoting 2j2*j-1 occurrences of width 3 in the width pattern of SRS(m):

CROSSREFS
STATUS

proposed

editing

Discussion
Sun Nov 03
17:45
Omar E. Pol: Minor edits.
#4 by Hartmut F. W. Hoft at Sun Nov 03 15:13:53 EST 2024
STATUS

editing

proposed

#3 by Hartmut F. W. Hoft at Sun Nov 03 15:13:47 EST 2024
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 *)

KEYWORD

nonn,changed,more

#2 by Hartmut F. W. Hoft at Sun Nov 03 15:06:57 EST 2024
NAME

allocated for Hartmut F. W. Hoft

Square Array read by antidiagonals: 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 2i-1 parts and has 2j-1 occurrences of maximum width 3 in its width pattern (row m of A341969).

DATA

72, 2450, 648, 1225, 120050, 450, 3969, 581042, 211250, 20808, 9801, 30625

OFFSET

1,1

COMMENTS

This sequence is a subsequence of A376829.

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.

EXAMPLE

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 2i-1 parts in SRS(m) and columns j denoting 2j-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

...

KEYWORD

allocated

nonn

AUTHOR

Hartmut F. W. Hoft, Nov 03 2024

STATUS

approved

editing

#1 by Hartmut F. W. Hoft at Sun Nov 03 15:06:57 EST 2024
NAME

allocated for Hartmut F. W. Hoft

KEYWORD

allocated

STATUS

approved