A334593 Number of ones in XOR-triangle with first row generated from the binary expansion of n.

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

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.

Records occur at 1, 2, 4, 5, 9, 11, 17, 18, 22, 35, 45, 69, 71, 73, 91, 139, 142, 146, 182, ...

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)) - A334592(n).

Example

For n = 53, a(53) = 12 because 53 = 110101_2 in binary, and the corresponding XOR-triangle has 12 ones:
  1 1 0 1 0 1
   0 1 1 1 1
    1 0 0 0
     1 0 0
      1 0
       1

Mathematica

Array[Total@ Flatten@ NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &] &, 74] (* Michael De Vlieger, May 08 2020 *)

Prog

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

Crossrefs

Cf. A000217, A070939.

Cf. A334556, A334591, A334592, A334594, A334595, A334596.

Sequence in context: A339695 A326846 A243220 * A293596 A301977 A085430

Adjacent sequences: A334590 A334591 A334592 * A334594 A334595 A334596

Keyword

nonn,base

Author

Peter Kagey, May 07 2020

Status

approved