A141803 Triangle read by rows derived from generalized Thue-Morse sequences.

1, 1, 1, 1, 2, 0, 1, 2, 1, 1, 1, 2, 3, 2, 0, 1, 2, 3, 1, 0, 0, 1, 2, 3, 4, 2, 2, 1, 1, 2, 3, 4, 1, 3, 0, 1, 1, 2, 3, 4, 5, 2, 0, 1, 0, 1, 2, 3, 4, 5, 1, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 2, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 1, 3, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 2, 4, 2, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7, 1, 3, 5, 3, 3, 0, 1

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, Jul 06 2008

Keywords

• nonn
• tabl

Comments

Triangle read by rows, antidiagonals of an array composed of generalized Thue-Morse sequences [defined in A010060, comment of Zizka]. For each row of the array, n>0; n-th term of m-th row (m>0) = sum of digits of n in base (m+1), mod (m+1).

Every row of the array starting from the n-th one as well as every row of the triangle starting from the (2*n-1)-th one begins from (1,2,3,...,n).

Row sums = A141804: (1, 2, 3, 5, 8, 7, 15, 15, 18, 22,...).

Row 1 of the array (corresponding to base 2) = A010060 (n>0), rows 2 - 8 are the sequences A053838 - A053844, row 9 = A053837.

Examples

			First few rows of the array are:
1, 1, 0, 1, 0, 0, 1, 1,...
1, 2, 1, 2, 0, 2, 0, 1,...
1, 2, 3, 1, 2, 3, 0, 2,...
1, 2, 3, 4, 1, 2, 3, 4,...
1, 2, 3, 4, 5, 1, 2, 3,...
1, 2, 3, 4, 5, 6, 1, 2,...
...
Triangle = antidiagonals of the array:
1;
1, 1;
1, 2, 0;
1, 2, 1, 1;
1, 2, 3, 2, 0;
1, 2, 3, 1, 0, 0;
1, 2, 3, 4, 2, 2, 1;
1, 2, 3, 4, 1, 3, 0, 1;
1, 2, 3, 4, 5, 2, 0, 1, 0;
1, 2, 3, 4, 5, 1, 3, 2, 1, 0;
1, 2, 3, 4, 5, 6, 2, 4, 3, 2, 1;
1, 2, 3, 4, 5, 6, 1, 3, 0, 0, 0, 0;
1, 2, 3, 4, 5, 6, 7, 2, 4, 2, 1, 2, 1;
1, 2, 3, 4, 5, 6, 7, 1, 3, 5, 3, 3, 0, 1;
...
a(8) = 2, = (3,2) of the array indicating that in the sequence 1,2,3,...mod 4, sum of digits of "2" mod 4 = 2.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..8001

Crossrefs

Cf. A010060, A053837 - A053844, A141804.

Programs

Mathematica

Flatten@Table[Mod[Total@IntegerDigits[n - i, i], i], {n, 16}, {i, n - 1, 2, -1}] (* Ivan Neretin, Jun 18 2018 *)

Extensions

Explanation in the Comments section corrected by Andrey Zabolotskiy, May 18 2016