A394648 Rectangular array, read by descending antidiagonals: the offset sum array of the Thue-Morse sequence, A010060; see Comments.

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, 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, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..77.

Formula

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

Example

Corner:
0   1   1   0   1   0   0   1   1   0   0
2   2   1   2   1   1   2   2   1   1   2
2   1   2   1   1   2   2   1   1   2   1
0   1   0   0   1   1   0   0   1   0   1
2   1   1   2   2   1   1   2   1   2   2
0   0   1   1   0   0   1   0   1   1   0
0   1   1   0   0   1   0   1   1   0   1
2   2   1   1   2   1   2   2   1   2   1
2   1   1   2   1   2   2   1   2   1   1
0   0   1   0   1   1   0   1   0   0   1
0   1   0   1   1   0   1   0   0   1   0

Mathematica

t[n_] := ThueMorse[Range[0, 200]][[n + 1]];
Table[t[n], {n, 0, 25}]
r[n_, k_] := t[n] + t[n + k];
u = Table[r[n, k], {n, 0, 15}, {k, 0, 20}]
Grid[u]

Crossrefs

Cf. A010060, A394649.

Sequence in context: A174950 A159906 A263444 * A276321 A152196 A024375

Adjacent sequences: A394645 A394646 A394647 * A394649 A394650 A394651

Keyword

nonn

Author

Clark Kimberling, Mar 27 2026

Status

approved