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A333624 Irregular triangle read by rows: T(n,k) = number of triangles of zeros with side length k in the XOR-triangle with first row generated from the binary expansion of n.

%I A333624 #20 Jul 29 2020 12:42:49
%S A333624 0,1,1,0,1,2,2,0,1,0,0,1,1,1,2,1,3,2,1,3,1,1,0,0,1,0,0,0,1,1,0,1,3,1,
%T A333624 1,2,1,2,2,0,1,5,3,1,2,0,1,1,2,3,1,5,1,2,3,1,1,0,1,0,0,0,1,0,0,0,0,1,
%U A333624 1,0,0,1,2,1,1,3,0,1,3,2,3,2,2,1,1,0,3
%N A333624 Irregular triangle read by rows: T(n,k) = number of triangles of zeros with side length k in the XOR-triangle with first row generated from the binary expansion of n.
%C A333624 An XOR-triangle is an inverted 0-1 triangle formed by choosing a top row and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(n) applied recursively until we reach a single bit.
%C A333624 Let b(n) = n written in binary and let L(n) = 1 + floor(log_2(n)) = A070939(n). Let => be a single iteration of XOR across pairs of bits in b(n). Let t(n) be the XOR triangle initiated by b(n). Thus we may refer to any bit in t(n) by the address S(i,j) with 1 <= i <= L(n) and 1 <= j <= L(n) - j + 1.
%C A333624 We detect triangles of zeros, which are "voids" amid surrounding 1's or undefined "space" in the t(n) via run lengths of -1 in S(i,j) - S(i-1,j) for i > 1, and for i = 1, run lengths of zeros.
%C A333624 A334591(n) = length of row n.
%C A333624 From _Michael De Vlieger_, May 27 2020: (Start)
%C A333624 We can compactify row n by taking the product of prime(k)^T(n,k) for 1 <= k <= A334591(n), decoding the compactified row using A067255. This way, we can compactify the populations of zero-triangles for large n. Example: for n = 151, t(151) has 3 singleton zeros and 4 zero-triangles of side length k = 2. Thus row 151 has {3, 4}. 2^3 * 3^4 = 8 * 81 = 648. A067255(648) = {3, 4}.
%C A333624 A333625(m) = Product(prime(k)^T(m,k)) for m in A334556 (rotationally symmetrical XOR-triangles).
%C A333624 A334896(m) = Product(prime(k)^T(m,k)) for m in A334769 (rotationally symmetrical XOR-triangles with central zero-triangles).
%C A333624 (End)
%H A333624 Michael De Vlieger, <a href="/A333624/b333624.txt">Table of n, a(n) for n = 1..10086</a> (rows 1 <= n <= 2500, flattened.)
%H A333624 Michael De Vlieger, <a href="/A333624/a333624.png">Montage</a> of XOR-triangle diagrams for 2 <= n <= 1025, with n in A334556 shown in black.
%H A333624 Michael De Vlieger, <a href="/A333624/a333624.txt">Table of rows 1 <= n <= 10000</a>, with terms k space-delimited, rows newline delimited.
%H A333624 Michael De Vlieger, <a href="http://vincico.com/seq/a334769.html">Central zero-triangles in rotationally symmetrical XOR-Triangles</a>, 2020.
%H A333624 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A333624 <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>
%e A333624 Table begins:
%e A333624 0;
%e A333624 1;
%e A333624 1;
%e A333624 0, 1;
%e A333624 2;
%e A333624 2;
%e A333624 0, 1;
%e A333624 0, 0, 1;
%e A333624 1, 1;
%e A333624 2, 1;
%e A333624 3;
%e A333624 2, 1;
%e A333624 3;
%e A333624 1, 1;
%e A333624 0, 0, 1;
%e A333624 0, 0, 0, 1;
%e A333624 1, 0, 1;
%e A333624 3, 1;
%e A333624 1, 2;
%e A333624 1, 2;
%e A333624 2, 0, 1;
%e A333624 ...
%e A333624 Let b(n) = n written in binary. Let => be a single iteration of XOR across pairs of bits in b(n). Let t(n) be the XOR triangle initiated by b(n).
%e A333624 Row 1 contains {0}, since b(1) = 1. Since the XOR triangle that results from a single 1-bit merely consists of that bit and since there are no zeros in the triangle t(1), we write the single term zero in this row.
%e A333624 Row 5 = {2} since b(5) = 101 => 11 => 0. Here we have 2 lone zeros, thus {2}.
%e A333624 Row 12 = {2, 1} since b(12) = 1100 => 010 => 11 => 0. We have 2 isolated zeros and 1 triangle of zeros with side length 2, thus {2, 1}.
%t A333624 Array[Function[w, If[Length@ # == 0, {0}, ReplacePart[ConstantArray[0, Max@ #[[All, 1]]], Map[#1 -> #2 & @@ # &, #]]] /. -Infinity -> 0 &@ Tally@ Flatten@ Array[If[# == 1, Map[If[First@ # == 1, Nothing, Length@ #] &, Split@ w[[#]] ], Map[If[First@ # == -1, Length@ #, Nothing] &, Split[w[[#]] - Most@ w[[# - 1]] ] ]] &, Length@ w]]@ NestWhileList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, IntegerDigits[#, 2], Length@ # > 1 &] &, 39] // Flatten
%Y A333624 Cf. A038554, A070939, A334591, A333625, A334896.
%K A333624 nonn,tabf
%O A333624 1,6
%A A333624 _Michael De Vlieger_, May 08 2020