A334595 Binary interpretation of the right diagonal of the XOR-triangle with first row generated from the binary expansion of n.

1, 1, 2, 1, 6, 3, 4, 1, 14, 4, 11, 2, 13, 7, 8, 1, 30, 11, 20, 7, 24, 13, 18, 3, 28, 9, 22, 5, 26, 15, 16, 1, 62, 20, 43, 13, 50, 24, 39, 5, 58, 16, 47, 9, 54, 28, 35, 2, 61, 23, 40, 14, 49, 27, 36, 6, 57, 19, 44, 10, 53, 31, 32, 1, 126, 43, 84, 24, 103, 50

Offset: 1

Author

Peter Kagey, May 07 2020

Keywords

• nonn
• base
• 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.

a(n) = n if and only if n is in A334556.

Conjecture: Records occur at 1 and at 2^n + 1.

Conjecture: a(n) = 1 if and only if n is a power of two.

Examples

			For n = 19, the binary expansion of 19 is 10011_2, and the XOR-triangle with first row generated from the binary expansion of 19 is:
  1 0 0 1 1
   1 0 1 0
    1 1 1
     0 0
      0
Reading the right side of the triangle starting from the upper-right corner gives 10100 which is the binary representation of 20 = a(19).
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..8191 (values less than 2^13)
• MathOverflow user DSM, Number triangle
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A334556, A334591, A334592, A334593, A334594, A334596.

Programs

PARI

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