A274912 Square array read by antidiagonals upwards in which each new term is the least nonnegative integer distinct from its neighbors.

0, 1, 2, 0, 3, 0, 1, 2, 1, 2, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

Author

Omar E. Pol, Jul 11 2016

Keywords

• nonn
• tabl

Comments

In the square array we have that:

Antidiagonal sums give A168237.

Odd-indexed rows give A010673.

Even-indexed rows give A010684.

Odd-indexed columns give A000035.

Even-indexed columns give A010693.

Odd-indexed antidiagonals give the initial terms of A010674.

Even-indexed antidiagonals give the initial terms of A000034.

Main diagonal gives A010674.

This is also a triangle read by rows in which each new term is the least nonnegative integer distinct from its neighbors.

In the triangle we have that:

Row sums give A168237.

Odd-indexed columns give A000035.

Even-indexed columns give A010693.

Odd-indexed diagonals give A010673.

Even-indexed diagonals give A010684.

Odd-indexed rows give the initial terms of A010674.

Even-indexed rows give the initial terms of A000034.

Odd-indexed antidiagonals give the initial terms of A010673.

Even-indexed antidiagonals give the initial terms of A010684.

Examples

			The corner of the square array begins:
0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, ...
0, 2, 0, 2, ...
1, 3, 1, ...
0, 2, ...
1, ...
...
The sequence written as a triangle begins:
0;
1, 2;
0, 3, 0;
1, 2, 1, 2;
0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2, 1, 2;
...
		

Crossrefs

Cf. A000034, A000035, A001477, A010673, A010674, A010684, A010693, A168237, A274913, A274920.

Programs

Maple

ListTools:-Flatten([seq([[0,3]$i,0,[1,2]$(i+1)],i=0..10)]); # Robert Israel, Nov 14 2016

Mathematica

Table[Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

Formula

a(n) = A274913(n) - 1.

From Robert Israel, Nov 14 2016: (Start)

G.f.: 3*x/(1-x^2) - Sum_{k>=0} (2*x^(2*k^2+3*k+1)-x^(2*k^2+5*k+3))/(1+x).

G.f. as triangle: x*(1+2*y+3*x*y)/((1-x^2*y^2)*(1-x^2)). (End)