cp's OEIS Frontend

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

%I A334591 #32 May 09 2020 00:50:04
%S A334591 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,
%T A334591 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,
%U A334591 3,3,3,3,2,3,2,4,3,2,3,4,3,2,2,4,5,3,3
%N A334591 Side length of largest triangle of zeros in the XOR-triangle with first row generated from the binary expansion of n.
%C A334591 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.
%C A334591 Records occur at a(2^n) = n.
%C A334591 Ones occur at 2, 3, 5, 6, 11, 13, 22, 27, 45, 54, 91, 109, 182, 219, 365, 438, 731, 877, 1462,...
%C A334591 a(n) <= A087117(n).
%H A334591 Peter Kagey, <a href="/A334591/b334591.txt">Table of n, a(n) for n = 1..8191</a>
%H A334591 MathOverflow user DSM, <a href="https://mathoverflow.net/q/359138/104733">Number triangle</a>
%H A334591 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A334591 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):
%e A334591   1 1 0 1 0 1
%e A334591    0 1 1 1 1
%e A334591     1 0 0 0
%e A334591      1 0 0
%e A334591       1 0
%e A334591        1
%t A334591 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 *)
%Y A334591 Cf. A087117, A334556, A334592, A334593, A334594, A334595, A334596.
%K A334591 nonn,base
%O A334591 1,4
%A A334591 _Peter Kagey_, May 07 2020