A201629 -id:A201629 - cp's OEIS frontend

A257857 Sequentially filled binary triangle rotated 180 degrees and then superimposed and added to the original triangle.

2, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 0, 2

Offset: 1

Craig Knecht, Jul 12 2015

The integers in the LINKS illustration hang like ornaments on a tree.

			Triangle T(n,k) begins:       Row sums
2;                                2
1,  1;                            2
0,  2,  0;                        2
1,  1,  1,  1;                    4
2,  0,  2,  0,  2;                6
1,  1,  1,  1,  1,  1;            6
0,  2,  0,  2,  0,  2,  0;        6
1,  1,  1,  1,  1,  1,  1,  1;    8

Craig Knecht, binary triangle rotated 180 degrees and superimposed on itself
Craig Knecht, Color coded contributions of the individual triangles

For row sums for the three other variations of this build process, see A186421, A201629, A240828.

A257857 := proc(n,k)
    if type(n,'even') then
        1 ;
    elif type((n+1)/2+k,'even') then
        2 ;
    else
        0;
    end if;
end proc:

T(n,k)=1 if n even, 1<=k<=n.
T(n,k)=2 if n odd and (n+1)/2+k even, 1<=k<=n.
T(n,k)=0 if n odd and (n+1)/2+k odd, 1<=k<=n.

A287797 Triangle read by rows: T(n,k) gives the independence number of the k X n knight graph.

Original entry on oeis.org

1, 2, 4, 3, 4, 5, 4, 4, 6, 8, 5, 6, 8, 10, 13, 6, 8, 9, 12, 15, 18, 7, 8, 11, 14, 18, 21, 25, 8, 8, 12, 16, 20, 24, 28, 32, 9, 10, 14, 18, 23, 27, 32, 36, 41, 10, 12, 15, 20, 25, 30, 35, 40, 45, 50, 11, 12, 17, 22, 28, 33, 39, 44, 50, 55, 61

Offset: 1

Eric W. Weisstein, Jun 01 2017

			1;
2, 4;
3, 4, 5;
4, 4, 6, 8;
5, 6, 8, 10, 13;
6, 8, 9, 12, 15, 18;

Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Knight Graph

Cf. A030978 (n X n knight graphs).

Cf. A201629 (2 X n knight graphs).

Table[IndependenceNumber[KnightTourGraph[m, n]], {n, 10}, {m, n}] // Flatten
Table[Piecewise[{{Max[m, n], Min[m, n] == 1}, {Max[m, n] + 1, Min[m, n] == 2 && Mod[Max[m, n], 2] == 1}, {4 Round[(Max[m, n] + 1)/4], Min[m, n] == 2 && Mod[Max[m, n], 2] == 0}, {m n/2, Mod[m n, 2] == 0}, {(m n + 1)/2, Mod[m n, 2] == 1}}], {n, 10}, {m, n}] // Flatten

T(n,n) = A030978(n).

T(n,2) = A201629(n+1).

cp's OEIS Frontend

A257857 Sequentially filled binary triangle rotated 180 degrees and then superimposed and added to the original triangle.

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