A334592 Number of zeros in XOR-triangle with first row generated from the binary expansion of n.

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

Offset

1,4

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.

Conjecture: Records occur at powers of two.

Links

Peter Kagey, Table of n, a(n) for n = 1..8191

MathOverflow user DSM, Number triangle

Index entries for sequences related to binary expansion of n

Formula

a(n) = A000217(A070939(n)) - A334593(n).

Example

For n = 53, a(53) = 9 because 53 = 110101_2 in binary, and the corresponding XOR-triangle has 9 zeros:
  1 1 0 1 0 1
   0 1 1 1 1
    1 0 0 0
     1 0 0
      1 0
       1

Mathematica

Array[Count[Flatten@ NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &], 0] &, 77] (* Michael De Vlieger, May 08 2020 *)

Prog

(PARI) a(n) = {my(b=binary(n), nb=#b-hammingweight(n)); for (n=1, #b-1, b = vector(#b-1, k, bitxor(b[k], b[k+1])); nb += #b-vecsum(b); ); nb; } \\ Michel Marcus, May 08 2020

Crossrefs

Cf. A000217, A070939.

Cf. also A334556, A334591, A334593, A334594, A334595, A334596.

Sequence in context: A335401 A324465 A361565 * A248756 A059942 A032450

Adjacent sequences: A334589 A334590 A334591 * A334593 A334594 A334595

Keyword

nonn,base

Author

Peter Kagey, May 07 2020

Status

approved