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A262057 Array based on the Stanley sequence S(0), A005836, by antidiagonals.

%I A262057 #54 Jan 06 2024 12:07:00
%S A262057 0,2,1,7,5,3,21,8,6,4,23,22,16,11,9,64,26,24,17,14,10,69,65,50,25,19,
%T A262057 15,12,71,70,67,53,48,20,18,13,193,80,78,68,59,49,34,29,27,207,194,
%U A262057 152,79,73,62,51,35,32,28,209,208,196,161,150,74,63,52,43,33,30
%N A262057 Array based on the Stanley sequence S(0), A005836, by antidiagonals.
%C A262057 This array is similar to a dispersion in that the first column is the minimal nonnegative sequence that contains no 3-term arithmetic progression, and each next column is the minimal sequence consisting of the numbers rejected from the previous column that contains no 3-term arithmetic progression.
%C A262057 A100480(n) describes which column n is sorted into.
%C A262057 The columns of the array form the greedy partition of the nonnegative integers into sequences that contain no 3-term arithmetic progression. - _Robert Israel_, Feb 03 2016
%H A262057 Max Barrentine and Robert Israel, <a href="/A262057/b262057.txt">Table of n, a(n) for n = 1..10011</a> (first 141 antidiagonals, flattened; n=1..77 from Max Barrentine)
%F A262057 A006997(A(n, k)) = k - 1. - _Rémy Sigrist_, Jan 06 2024
%e A262057 From the top-left corner, this array starts:
%e A262057    0   2   7  21  23  64
%e A262057    1   5   8  22  26  65
%e A262057    3   6  16  24  50  67
%e A262057    4  11  17  25  53  68
%e A262057    9  14  19  48  59  73
%e A262057   10  15  20  49  62  74
%p A262057 M:= 20: # to get the first M antidiagonals
%p A262057 for i from 1 to M do B[i]:= {}: F[i]:= {}: od:
%p A262057 countdowns:= Vector(M,j->M+1-j):
%p A262057 for x from 0 while max(countdowns) > 0 do
%p A262057   for i from 1 do
%p A262057      if not member(x, F[i]) then
%p A262057        F[i]:= F[i] union map(y -> 2*x-y, B[i]);
%p A262057        B[i]:= B[i] union {x};
%p A262057        countdowns[i]:= countdowns[i] - 1;
%p A262057      break
%p A262057     fi
%p A262057   od;
%p A262057 od:
%p A262057 seq(seq(B[n+1-i][i], i=1..n),n=1..M); # _Robert Israel_, Feb 03 2016
%o A262057 (MATLAB)
%o A262057 function  A = A262057( M, N )
%o A262057 % to get first M antidiagonals using x up to N
%o A262057 B = cell(1,M);
%o A262057 F = zeros(M,N+1);
%o A262057 countdowns = [M:-1:1];
%o A262057 for x=0:N
%o A262057     if max(countdowns) == 0
%o A262057         break
%o A262057     end
%o A262057     for i=1:M
%o A262057         if F(i,x+1) == 0
%o A262057             newforb = 2*x - B{i};
%o A262057             newforb = newforb(newforb <= N & newforb >= 1);
%o A262057             F(i,newforb+1) = 1;
%o A262057             B{i}(end+1) = x;
%o A262057             countdowns(i) = countdowns(i)-1;
%o A262057             break
%o A262057         end
%o A262057     end
%o A262057 end
%o A262057 if max(countdowns) > 0
%o A262057     [~,jmax] = max(countdowns);
%o A262057     jmax = jmax(1);
%o A262057     error ('Need larger N: B{%d} has only %d elements',jmax,numel(B{jmax}));
%o A262057 end
%o A262057 A = zeros(1,M*(M+1)/2);
%o A262057 k = 0;
%o A262057 for n=1:M
%o A262057     for i=1:n
%o A262057         k=k+1;
%o A262057         A(k) = B{n+1-i}(i);
%o A262057     end
%o A262057 end
%o A262057 end % _Robert Israel_, Feb 03 2016
%Y A262057 First column is A005836.
%Y A262057 First row is A265316.
%Y A262057 Cf. A006997, A074940, A100480.
%K A262057 nonn,tabl
%O A262057 1,2
%A A262057 _Max Barrentine_, Nov 29 2015