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

1,2

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.

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A038554, A070939.

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

Sequence in context: A278529 A160588 A081171 * A359336 A062778 A363620

Adjacent sequences: A334591 A334592 A334593 * A334595 A334596 A334597

Keyword

nonn,base,tabf,look

Author

Peter Kagey, May 07 2020

Status

approved