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Square array read by antidiagonals upwards in which each new term is the least nonnegative integer distinct from its neighbors.
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%I #22 Nov 14 2016 17:24:15

%S 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,

%T 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,

%U 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

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

%C In the square array we have that:

%C Antidiagonal sums give A168237.

%C Odd-indexed rows give A010673.

%C Even-indexed rows give A010684.

%C Odd-indexed columns give A000035.

%C Even-indexed columns give A010693.

%C Odd-indexed antidiagonals give the initial terms of A010674.

%C Even-indexed antidiagonals give the initial terms of A000034.

%C Main diagonal gives A010674.

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

%C In the triangle we have that:

%C Row sums give A168237.

%C Odd-indexed columns give A000035.

%C Even-indexed columns give A010693.

%C Odd-indexed diagonals give A010673.

%C Even-indexed diagonals give A010684.

%C Odd-indexed rows give the initial terms of A010674.

%C Even-indexed rows give the initial terms of A000034.

%C Odd-indexed antidiagonals give the initial terms of A010673.

%C Even-indexed antidiagonals give the initial terms of A010684.

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

%F From _Robert Israel_, Nov 14 2016: (Start)

%F 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).

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

%e The corner of the square array begins:

%e 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ...

%e 1, 3, 1, 3, 1, 3, 1, 3, 1, ...

%e 0, 2, 0, 2, 0, 2, 0, 2, ...

%e 1, 3, 1, 3, 1, 3, 1, ...

%e 0, 2, 0, 2, 0, 2, ...

%e 1, 3, 1, 3, 1, ...

%e 0, 2, 0, 2, ...

%e 1, 3, 1, ...

%e 0, 2, ...

%e 1, ...

%e ...

%e The sequence written as a triangle begins:

%e 0;

%e 1, 2;

%e 0, 3, 0;

%e 1, 2, 1, 2;

%e 0, 3, 0, 3, 0;

%e 1, 2, 1, 2, 1, 2;

%e 0, 3, 0, 3, 0, 3, 0;

%e 1, 2, 1, 2, 1, 2, 1, 2;

%e 0, 3, 0, 3, 0, 3, 0, 3, 0;

%e 1, 2, 1, 2, 1, 2, 1, 2, 1, 2;

%e ...

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

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

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

%K nonn,tabl

%O 0,3

%A _Omar E. Pol_, Jul 11 2016