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%I #54 Jan 06 2024 12:07:00
%S 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 15,12,71,70,67,53,48,20,18,13,193,80,78,68,59,49,34,29,27,207,194,
%U 152,79,73,62,51,35,32,28,209,208,196,161,150,74,63,52,43,33,30
%N Array based on the Stanley sequence S(0), A005836, by antidiagonals.
%C 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 A100480(n) describes which column n is sorted into.
%C 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 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 A006997(A(n, k)) = k - 1. - _Rémy Sigrist_, Jan 06 2024
%e From the top-left corner, this array starts:
%e 0 2 7 21 23 64
%e 1 5 8 22 26 65
%e 3 6 16 24 50 67
%e 4 11 17 25 53 68
%e 9 14 19 48 59 73
%e 10 15 20 49 62 74
%p M:= 20: # to get the first M antidiagonals
%p for i from 1 to M do B[i]:= {}: F[i]:= {}: od:
%p countdowns:= Vector(M,j->M+1-j):
%p for x from 0 while max(countdowns) > 0 do
%p for i from 1 do
%p if not member(x, F[i]) then
%p F[i]:= F[i] union map(y -> 2*x-y, B[i]);
%p B[i]:= B[i] union {x};
%p countdowns[i]:= countdowns[i] - 1;
%p break
%p fi
%p od;
%p od:
%p seq(seq(B[n+1-i][i], i=1..n),n=1..M); # _Robert Israel_, Feb 03 2016
%o (MATLAB)
%o function A = A262057( M, N )
%o % to get first M antidiagonals using x up to N
%o B = cell(1,M);
%o F = zeros(M,N+1);
%o countdowns = [M:-1:1];
%o for x=0:N
%o if max(countdowns) == 0
%o break
%o end
%o for i=1:M
%o if F(i,x+1) == 0
%o newforb = 2*x - B{i};
%o newforb = newforb(newforb <= N & newforb >= 1);
%o F(i,newforb+1) = 1;
%o B{i}(end+1) = x;
%o countdowns(i) = countdowns(i)-1;
%o break
%o end
%o end
%o end
%o if max(countdowns) > 0
%o [~,jmax] = max(countdowns);
%o jmax = jmax(1);
%o error ('Need larger N: B{%d} has only %d elements',jmax,numel(B{jmax}));
%o end
%o A = zeros(1,M*(M+1)/2);
%o k = 0;
%o for n=1:M
%o for i=1:n
%o k=k+1;
%o A(k) = B{n+1-i}(i);
%o end
%o end
%o end % _Robert Israel_, Feb 03 2016
%Y First column is A005836.
%Y First row is A265316.
%Y Cf. A006997, A074940, A100480.
%K nonn,tabl
%O 1,2
%A _Max Barrentine_, Nov 29 2015
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