A394648 - OEIS

%I #6 Mar 31 2026 18:04:56

%S 0,1,2,1,2,2,0,1,1,0,1,2,2,1,2,0,1,1,0,1,0,0,1,1,0,1,0,0,1,2,2,1,2,1,

%T 1,2,1,2,2,1,2,1,1,2,2,0,1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,2,

%U 2,1,2,1,1,2,2,1,1,2

%N Rectangular array, read by descending antidiagonals: the offset sum array of the Thue-Morse sequence, A010060; see Comments.

%C The offset sum array of a sequence (s(n)) is introduced here as the rectangular array whose n-th row is s(n) + s(n+k), for n>= 0, k >= 0. Properties of the array when s=A010060: (1) each row has exactly 2 distinct terms; (2) no two rows are identical; (3) every column except the first has exactly 3 distinct terms; (4) no two columns are identical; (5) the diagonal (s(n)+s(n+1)) is A000012; (6) (column 1) = (main diagonal) = 2*A010060.

%F r(n,k) = s(n) + s(n+k), for n >= 0, k >= 0, where s = A010060.

%e Corner:

%e 0   1   1   0   1   0   0   1   1   0   0

%e 2   2   1   2   1   1   2   2   1   1   2

%e 2   1   2   1   1   2   2   1   1   2   1

%e 0   1   0   0   1   1   0   0   1   0   1

%e 2   1   1   2   2   1   1   2   1   2   2

%e 0   0   1   1   0   0   1   0   1   1   0

%e 0   1   1   0   0   1   0   1   1   0   1

%e 2   2   1   1   2   1   2   2   1   2   1

%e 2   1   1   2   1   2   2   1   2   1   1

%e 0   0   1   0   1   1   0   1   0   0   1

%e 0   1   0   1   1   0   1   0   0   1   0

%t t[n_] := ThueMorse[Range[0, 200]][[n + 1]];

%t Table[t[n], {n, 0, 25}]

%t r[n_, k_] := t[n] + t[n + k];

%t u = Table[r[n, k], {n, 0, 15}, {k, 0, 20}]

%t Grid[u]

%Y Cf. A010060, A394649.

%K nonn

%O 0,3

%A _Clark Kimberling_, Mar 27 2026