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

%I A334592 #19 May 09 2020 00:51:00
%S A334592 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,
%T A334592 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,
%U A334592 11,15,21,16,14,15,16,13,13,12,14,11,13,12,17,12
%N A334592 Number of zeros in XOR-triangle with first row generated from the binary expansion of n.
%C A334592 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 A334592 Conjecture: Records occur at powers of two.
%H A334592 Peter Kagey, <a href="/A334592/b334592.txt">Table of n, a(n) for n = 1..8191</a>
%H A334592 MathOverflow user DSM, <a href="https://mathoverflow.net/q/359138/104733">Number triangle</a>
%H A334592 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A334592 a(n) = A000217(A070939(n)) - A334593(n).
%e A334592 For n = 53, a(53) = 9 because 53 = 110101_2 in binary, and the corresponding XOR-triangle has 9 zeros:
%e A334592   1 1 0 1 0 1
%e A334592    0 1 1 1 1
%e A334592     1 0 0 0
%e A334592      1 0 0
%e A334592       1 0
%e A334592        1
%t A334592 Array[Count[Flatten@ NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &], 0] &, 77] (* _Michael De Vlieger_, May 08 2020 *)
%o A334592 (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
%Y A334592 Cf. A000217, A070939.
%Y A334592 Cf. also A334556, A334591, A334593, A334594, A334595, A334596.
%K A334592 nonn,base
%O A334592 1,4
%A A334592 _Peter Kagey_, May 07 2020