

A333624


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


4



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, 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, 1, 0, 0, 1, 2, 1, 1, 3, 0, 1, 3, 2, 3, 2, 2, 1, 1, 0, 3
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OFFSET

1,6


COMMENTS

An XORtriangle is an inverted 01 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.
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.
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(i1,j) for i > 1, and for i = 1, run lengths of zeros.
A334591(n) = length of row n.
From Michael De Vlieger, May 27 2020: (Start)
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 zerotriangles for large n. Example: for n = 151, t(151) has 3 singleton zeros and 4 zerotriangles of side length k = 2. Thus row 151 has {3, 4}. 2^3 * 3^4 = 8 * 81 = 648. A067255(648) = {3, 4}.
A333625(m) = Product(prime(k)^T(m,k)) for m in A334556 (rotationally symmetrical XORtriangles).
A334896(m) = Product(prime(k)^T(m,k)) for m in A334769 (rotationally symmetrical XORtriangles with central zerotriangles).
(End)


LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10086 (rows 1 <= n <= 2500, flattened.)
Michael De Vlieger, Montage of XORtriangle diagrams for 2 <= n <= 1025, with n in A334556 shown in black.
Michael De Vlieger, Table of rows 1 <= n <= 10000, with terms k spacedelimited, rows newline delimited.
Michael De Vlieger, Central zerotriangles in rotationally symmetrical XORTriangles, 2020.
Index entries for sequences related to binary expansion of n
Index entries for sequences related to XORtriangles


EXAMPLE

Table begins:
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;
1, 2;
1, 2;
2, 0, 1;
...
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).
Row 1 contains {0}, since b(1) = 1. Since the XOR triangle that results from a single 1bit 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.
Row 5 = {2} since b(5) = 101 => 11 => 0. Here we have 2 lone zeros, thus {2}.
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}.


MATHEMATICA

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


CROSSREFS

Cf. A038554, A070939, A334591, A333625, A334896.
Sequence in context: A248211 A049502 A242284 * A306595 A332996 A292592
Adjacent sequences: A333621 A333622 A333623 * A333625 A333626 A333627


KEYWORD

nonn,tabf


AUTHOR

Michael De Vlieger, May 08 2020


STATUS

approved



