A334594 Irregular table read by rows: T(n,k) is the binary interpretation of the k-th row of the XOR-triangle with first row generated from the binary expansion of n. 1 <= k <= A070939(n).

1, 2, 1, 3, 0, 4, 2, 1, 5, 3, 0, 6, 1, 1, 7, 0, 0, 8, 4, 2, 1, 9, 5, 3, 0, 10, 7, 0, 0, 11, 6, 1, 1, 12, 2, 3, 0, 13, 3, 2, 1, 14, 1, 1, 1, 15, 0, 0, 0, 16, 8, 4, 2, 1, 17, 9, 5, 3, 0, 18, 11, 6, 1, 1, 19, 10, 7, 0, 0, 20, 14, 1, 1, 1, 21, 15, 0, 0, 0

Offset: 1

Author

Peter Kagey, May 07 2020

Keywords

• nonn
• base
• tabf
• look

Comments

An XOR-triangle is an inverted 0-1 triangle formed by choosing a top row and having each entry in the subsequent rows be the XOR of the two values above it.

The second column is A038554.

Examples

			Table begins:
   1;
   2, 1;
   3, 0;
   4, 2, 1;
   5, 3, 0;
   6, 1, 1;
   7, 0, 0;
   8, 4, 2, 1;
   9, 5, 3, 0;
  10, 7, 0, 0;
  11, 6, 1, 1;
For the 11th row, the binary expansion of 11 is 1011_2, and the corresponding XOR-triangle is
  1 0 1 1
   1 1 0
    0 1
     1
Reading the rows of this triangle in binary gives 11, 6, 1, 1.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..9217 (first 1023 rows)
• MathOverflow user DSM, Number triangle
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A038554, A070939.

Cf. A334556, A334591, A334592, A334593, A334595, A334596.

Programs

Mathematica

Array[Prepend[FromDigits[#, 2] & /@ #2, #1] & @@ {#, Rest@ NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &]} &, 21] // Flatten (* Michael De Vlieger, May 08 2020 *)

PARI

row(n) = {my(b=binary(n), v=vector(#b)); v[1] = n; for (n=1, #b-1, b = vector(#b-1, k, bitxor(b[k], b[k+1])); v[n+1] = fromdigits(b, 2);); v;} \\ Michel Marcus, May 08 2020