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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.

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(i-1,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 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}.

A333625(m) = Product(prime(k)^T(m,k)) for m in A334556 (rotationally symmetrical XOR-triangles).

A334896(m) = Product(prime(k)^T(m,k)) for m in A334769 (rotationally symmetrical XOR-triangles with central zero-triangles).

(End)

This sequence indexes the smallest number m = A334769(k) that, when expressed in binary b(k), generates a rotationally symmetrical XOR-triangle (RST) that features a central zero-triangle (CZT) with frame width n.

A "frame width" is the number of iterations j required to generate the first run of zeros in a CZT of an RST.

Let L = A070939(m) for m in A334769. For RSTs, j > 1, since a solid run of L 1s given a recursive XOR function applied to each pair of adjacent bits, would give rise to a solid run of (L - 1) zeros in the next iteration, and every iteration thereafter consists of zeros. Therefore m = (2(L - 1) - 1) is not rotationally symmetrical except when L = 1.

Sequence A334556 lists numbers m that produce RSTs; A334769 those RSTs that feature CZTs. Sequence A334796 renders the frame widths j for numbers in A334769.

For n = 7, A070939(a(n)) > 3(7) + 1 = 22, but is likely much larger, given a(6). a(7) is likely a number with more than 40 bits.

We can devise a similar sequence for any fixed base b >= 2; the present sequence corresponds to b = 3, and A334556 corresponds to b = 2.

This sequence is infinite as it contains A048328.

If k belongs to the sequence, then A004488(k) and A030102(k) belong to the sequence.

Empirically, there are 2*3^floor((w-1)/3) positive terms with w ternary digits.

For any k, if t appears above u and v in T_k, then t + u + v = 0 (mod 3) and #{t, u, v} = 1 or 3 (the three values are either equal or all distinct); each value is uniquely determined by the two others in the same way: t = (-u-v) mod 3, u = (-t-v) mod 3, v = (-t-u) mod 3; this means that we can reconstruct T_k from any of its three sides.

If some row of T_k, say r, has w values and corresponds to the ternary expansion of m, then the row above r corresponds to the w-1 rightmost digits of the ternary expansion of A060587(m).

All positive terms belong to A297250 (their most significant digit equals their least significant digit in base 3).

An XOR-triangle T(m) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n and having each entry in subsequent rows be the XOR of the two values above it, i.e., A038554(m) applied recursively until we reach a single bit.

A334556 is the sequence of rotationally symmetrical T(m).

A central zero-triangle (CZT) is a field of contiguous 0-bits with side length n in T(m) surrounded on all sides by a layer of 1 bits, and generally k > 1 bits of any parity. Alternatively, these might be referred to as "central bubbles".

Subset of A334556.

No zero appears in the center of the figure, thus a(n) does not intersect A334769.

Numbers m with A070939(m) (mod 3) = 1 involving alternating run lengths of a singleton zero separated by a pair of 1s in the binary expansion, admitting an initial or final singleton 1.

Subset of A334769 which is a subset of A334556.

Numbers m in this sequence A070939(m) (mod 3) = 2. The numbers in this sequence can be constructed using run lengths of bits.

2n has the reverse run length pattern as 2n - 1. a(1) has the run lengths {1, 2, 1, 1, 3}, while a(2) has {3, 1, 1, 2, 1}, etc.

For n = 1 (mod 8): 12..(1132)..113;

For n = 3 (mod 8): 113..(2113)..2112;

For n = 5 (mod 8): 11123..(1123)..1122;

For n = 7 (mod 8): 123112..(3112)..31123, where the parenthetic run lengths occur, when they occur, in multiples of 3. Thus, a(9) has the run length form 12113211321132113 = binary 10010111001011100101110010111 = decimal 317049751.

Subset of A334769 which is a subset of A334556.

Numbers m in this sequence A070939(m) (mod 3) = 0. All m have first and last bits = 1.

The numbers in this sequence can be constructed using run lengths of bits thus: 12..(42)..3 or the reverse 3..(24)..21, with at least one copy of the pair of parenthetic numbers.

Thus, the smallest number m has run lengths {1, 2, 4, 2, 3}, which is the binary 100111100111 = decimal 2535.

2n has the reverse run length pattern as 2n - 1. a(3) has the run lengths {1, 2, 4, 2, 4, 2, 3}, while a(4) has {3, 2, 4, 2, 4, 2, 1}, etc.

Row a(n) of A067255 = row A334769(n) of A333624.

An XOR-triangle t(n) is an inverted 0-1 triangle formed by choosing a top row the binary rendition of n 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 T(n,k) address the terms in the k-th position of row n in A333624.

This sequence encodes T(n,k) via A067255 to succinctly express the number of zero-triangles in A334769(n). To decode a(n) => A333624(A334769(n)), we use A067255(a(n)).

This sequence is a variant of A334556 and A361818.

This sequence is infinite as it contains A003462.

Empirically, for any w > 0, there are A127975(w-1) terms with w balanced ternary digits (ignoring leading zeros).

If k is a term then A338246(k) is also a term.

There are three possible axes of symmetry:

.

. V

. U W

. .___._____.

. \ . . /

. \ . /

. . .

. . \ . / .

. W \ / U

. .

.

. V

.

- symmetry through axis U-U is only possible for the numbers 0 and 1,

- symmetry through axis V-V corresponds to binary palindromes,

- symmetry through axis W-W corresponds to number k such that A334727(k) is a binary palindrome,

- 0 and 1 are the only terms whose XOR-triangles have the three symmetries,

- XOR-triangles of other terms have only one kind of symmetry.