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

%I A274912 #22 Nov 14 2016 17:24:15
%S A274912 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 A274912 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 A274912 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 A274912 Square array read by antidiagonals upwards in which each new term is the least nonnegative integer distinct from its neighbors.
%C A274912 In the square array we have that:
%C A274912 Antidiagonal sums give A168237.
%C A274912 Odd-indexed rows give A010673.
%C A274912 Even-indexed rows give A010684.
%C A274912 Odd-indexed columns give A000035.
%C A274912 Even-indexed columns give A010693.
%C A274912 Odd-indexed antidiagonals give the initial terms of A010674.
%C A274912 Even-indexed antidiagonals give the initial terms of A000034.
%C A274912 Main diagonal gives A010674.
%C A274912 This is also a triangle read by rows in which each new term is the least nonnegative integer distinct from its neighbors.
%C A274912 In the triangle we have that:
%C A274912 Row sums give A168237.
%C A274912 Odd-indexed columns give A000035.
%C A274912 Even-indexed columns give A010693.
%C A274912 Odd-indexed diagonals give A010673.
%C A274912 Even-indexed diagonals give A010684.
%C A274912 Odd-indexed rows give the initial terms of A010674.
%C A274912 Even-indexed rows give the initial terms of A000034.
%C A274912 Odd-indexed antidiagonals give the initial terms of A010673.
%C A274912 Even-indexed antidiagonals give the initial terms of A010684.
%F A274912 a(n) = A274913(n) - 1.
%F A274912 From _Robert Israel_, Nov 14 2016: (Start)
%F A274912 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 A274912 G.f. as triangle: x*(1+2*y+3*x*y)/((1-x^2*y^2)*(1-x^2)). (End)
%e A274912 The corner of the square array begins:
%e A274912 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ...
%e A274912 1, 3, 1, 3, 1, 3, 1, 3, 1, ...
%e A274912 0, 2, 0, 2, 0, 2, 0, 2, ...
%e A274912 1, 3, 1, 3, 1, 3, 1, ...
%e A274912 0, 2, 0, 2, 0, 2, ...
%e A274912 1, 3, 1, 3, 1, ...
%e A274912 0, 2, 0, 2, ...
%e A274912 1, 3, 1, ...
%e A274912 0, 2, ...
%e A274912 1, ...
%e A274912 ...
%e A274912 The sequence written as a triangle begins:
%e A274912 0;
%e A274912 1, 2;
%e A274912 0, 3, 0;
%e A274912 1, 2, 1, 2;
%e A274912 0, 3, 0, 3, 0;
%e A274912 1, 2, 1, 2, 1, 2;
%e A274912 0, 3, 0, 3, 0, 3, 0;
%e A274912 1, 2, 1, 2, 1, 2, 1, 2;
%e A274912 0, 3, 0, 3, 0, 3, 0, 3, 0;
%e A274912 1, 2, 1, 2, 1, 2, 1, 2, 1, 2;
%e A274912 ...
%p A274912 ListTools:-Flatten([seq([[0,3]$i,0,[1,2]$(i+1)],i=0..10)]); # _Robert Israel_, Nov 14 2016
%t A274912 Table[Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* _Michael De Vlieger_, Nov 14 2016 *)
%Y A274912 Cf. A000034, A000035, A001477, A010673, A010674, A010684, A010693, A168237, A274913, A274920.
%K A274912 nonn,tabl
%O A274912 0,3
%A A274912 _Omar E. Pol_, Jul 11 2016