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

1,5

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.

Links

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

Example

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.

Mathematica

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

Crossrefs

Cf. A010060, A053837 - A053844, A141804.

Sequence in context: A033785 A033781 A337475 * A249147 A260648 A127242

Adjacent sequences: A141800 A141801 A141802 * A141804 A141805 A141806

Keyword

nonn,tabl

Author

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

Extensions

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

Status

approved