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A262057 Array based on the Stanley sequence S(0), A005836, by antidiagonals. 2
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, 15, 12, 71, 70, 67, 53, 48, 20, 18, 13, 193, 80, 78, 68, 59, 49, 34, 29, 27, 207, 194, 152, 79, 73, 62, 51, 35, 32, 28, 209, 208, 196, 161, 150, 74, 63, 52, 43, 33, 30 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

A100480(n) describes which column n is sorted into.

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

LINKS

Max Barrentine and Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 antidiagonals, flattened; n=1..77 from Max Barrentine)

EXAMPLE

From the top-left corner, this array starts:

   0   2   7  21  23  64

   1   5   8  22  26  65

   3   6  16  24  50  67

   4  11  17  25  53  68

   9  14  19  48  59  73

  10  15  20  49  62  74

MAPLE

M:= 20: # to get the first M antidiagonals

for i from 1 to M do B[i]:= {}: F[i]:= {}: od:

countdowns:= Vector(M, j->M+1-j):

for x from 0 while max(countdowns) > 0 do

  for i from 1 do

     if not member(x, F[i]) then

       F[i]:= F[i] union map(y -> 2*x-y, B[i]);

       B[i]:= B[i] union {x};

       countdowns[i]:= countdowns[i] - 1;

     break

    fi

  od;

od:

seq(seq(B[n+1-i][i], i=1..n), n=1..M); # Robert Israel, Feb 03 2016

PROG

(MATLAB)

function  A = A262057( M, N )

% to get first M antidiagonals using x up to N

B = cell(1, M);

F = zeros(M, N+1);

countdowns = [M:-1:1];

for x=0:N

    if max(countdowns) == 0

        break

    end

    for i=1:M

        if F(i, x+1) == 0

            newforb = 2*x - B{i};

            newforb = newforb(newforb <= N & newforb >= 1);

            F(i, newforb+1) = 1;

            B{i}(end+1) = x;

            countdowns(i) = countdowns(i)-1;

            break

        end

    end

end

if max(countdowns) > 0

    [~, jmax] = max(countdowns);

    jmax = jmax(1);

    error ('Need larger N: B{%d} has only %d elements', jmax, numel(B{jmax}));

end

A = zeros(1, M*(M+1)/2);

k = 0;

for n=1:M

    for i=1:n

        k=k+1;

        A(k) = B{n+1-i}(i);

    end

end

end % Robert Israel, Feb 03 2016

CROSSREFS

First column is A005836.

First row is A265316.

Cf. A074940, A100480.

Sequence in context: A124821 A104030 A206533 * A082791 A299238 A077230

Adjacent sequences:  A262054 A262055 A262056 * A262058 A262059 A262060

KEYWORD

nonn,tabl

AUTHOR

Max Barrentine, Nov 29 2015

STATUS

approved

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Last modified January 18 11:33 EST 2019. Contains 319271 sequences. (Running on oeis4.)