A334594 - OEIS

%I #23 May 09 2020 00:51:38

%S 1,2,1,3,0,4,2,1,5,3,0,6,1,1,7,0,0,8,4,2,1,9,5,3,0,10,7,0,0,11,6,1,1,

%T 12,2,3,0,13,3,2,1,14,1,1,1,15,0,0,0,16,8,4,2,1,17,9,5,3,0,18,11,6,1,

%U 1,19,10,7,0,0,20,14,1,1,1,21,15,0,0,0

%N Irregular table read by rows: T(n,k) is the binary interpretation of the k-th row of the XOR-triangle with first row generated from the binary expansion of n. 1 <= k <= A070939(n).

%C 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 The second column is A038554.

%H Peter Kagey, <a href="/A334594/b334594.txt">Table of n, a(n) for n = 1..9217</a> (first 1023 rows)

%H MathOverflow user DSM, <a href="https://mathoverflow.net/q/359138/104733">Number triangle</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%e Table begins:

%e    1;

%e    2, 1;

%e    3, 0;

%e    4, 2, 1;

%e    5, 3, 0;

%e    6, 1, 1;

%e    7, 0, 0;

%e    8, 4, 2, 1;

%e    9, 5, 3, 0;

%e   10, 7, 0, 0;

%e   11, 6, 1, 1;

%e For the 11th row, the binary expansion of 11 is 1011_2, and the corresponding XOR-triangle is

%e   1 0 1 1

%e    1 1 0

%e     0 1

%e      1

%e Reading the rows of this triangle in binary gives 11, 6, 1, 1.

%t Array[Prepend[FromDigits[#, 2] & /@ #2, #1] & @@ {#, Rest@ NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &]} &, 21] // Flatten (* _Michael De Vlieger_, May 08 2020 *)

%o (PARI) row(n) = {my(b=binary(n), v=vector(#b)); v[1] = n; for (n=1, #b-1, b = vector(#b-1, k, bitxor(b[k], b[k+1])); v[n+1] = fromdigits(b, 2);); v;} \\ _Michel Marcus_, May 08 2020

%Y Cf. A038554, A070939.

%Y Cf. A334556, A334591, A334592, A334593, A334595, A334596.

%K nonn,base,tabf,look

%O 1,2

%A _Peter Kagey_, May 07 2020