A334591 Side length of largest triangle of zeros in the XOR-triangle with first row generated from the binary expansion of n.

0, 1, 1, 2, 1, 1, 2, 3, 2, 2, 1, 2, 1, 2, 3, 4, 3, 2, 2, 2, 3, 1, 2, 3, 2, 2, 1, 2, 2, 3, 4, 5, 4, 3, 3, 2, 2, 3, 2, 3, 2, 4, 3, 2, 1, 2, 3, 4, 3, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 3, 3, 4, 5, 6, 5, 4, 4, 3, 3, 3, 3, 3, 2, 3, 2, 4, 3, 2, 3, 4, 3, 2, 2, 4, 5, 3, 3

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.

Records occur at a(2^n) = n.

Ones occur at 2, 3, 5, 6, 11, 13, 22, 27, 45, 54, 91, 109, 182, 219, 365, 438, 731, 877, 1462,...

a(n) <= A087117(n).

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

Example

For n = 53, a(53) = 3 because 53 = 110101_2 in binary, and the largest triangle of 0s in the corresponding XOR-triangle has size 3 (see third, fourth, and fifth rows):
  1 1 0 1 0 1
   0 1 1 1 1
    1 0 0 0
     1 0 0
      1 0
       1

Mathematica

Array[Function[w, Max@ Flatten@ Array[If[# == 1, If[First@ # == 1, Nothing, Length@ #] & /@ Split@ w[[#]], If[First@ # == -1, Length@ #, Nothing] & /@ Split[w[[#]] - Most@ w[[# - 1]] ] ] &, Length@ w] /. -Infinity -> 0]@ NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &] &, 105] (* Michael De Vlieger, May 08 2020 *)

Crossrefs

Cf. A087117, A334556, A334592, A334593, A334594, A334595, A334596.

Sequence in context: A338198 A091598 A144021 * A177962 A246552 A161091

Adjacent sequences: A334588 A334589 A334590 * A334592 A334593 A334594

Keyword

nonn,base

Author

Peter Kagey, May 07 2020

Status

approved