Aspects of XOR-triangles T(A334836(j)). Michael Thomas De Vlieger, St. Louis, Missouri. 2020 0520 1400. n is the side-length of the central zero-triangle (CZT) of the rotationally symmetrical XOR-triangle T(A334771(n)) generated by A334771(n). A334836(j) is the smallest number that produces a CZT with frame width j. Frame width j refers to the number of iterations of the XOR function until CZT zeros are generated. Example: The smallest number that produces a CZT of side length 3 is n = 2359. This is the 11th term in A334769 (list of rotationally symmetrical XOR-triangles that produce CZTs), and the 24th term in A334556 (list of rotationally symmetrical XOR-triangles). This XOR-triangle T(2359) has 9 ZTs of side length 1, 6 of side length 2, and 1 of side length 3, which corresponds with the CZT of side length 3. The diagram below illustrates T(10707), showing 0s as "." and 1s as "@" for clarity: @ . @ . . @ @ @ . @ . . @ @ @ @ @ . @ . . @ @ @ . @ . . . @ @ @ . @ . . @ @ @ . @ . . @ @ @ . @ . . @ @ . @ . . @ @ @ . . @ @ @ . @ . . @ @ . . @ @ @ . @ @ . @ . . @ @ @ @ @ . @ . . . @ @ @ . @ . . @ @ . . @ @ In this figure we see four rows between the central triangle's top row and empty space above the figure. Thus frame width j = 4. Legend: j index = frame width of central zero triangle. i index of n in A334769. m index of n in A334556. n seed number that, when converted to binary, is the smallest number that generates a rotationally symmetrical XOR-triangle with a central zero-triangle of side length k. n_16 n converted to hexadecimal. L floor(log_2 n) = 3j + k; binary integer length of n. k side length of CZT = A334770(n). Run Length Recipe We write the number of consecutive bits of same parity. Since the first bit must be 1, we interpret every odd run length as pertaining to 1, and evens to 0. Thus, run length recipe "145" would translate to 1000011111, which is 543 decimally. For some RSTs with CZTs, due to the size of the CZT, a modular pattern in run lengths arise, which can be abbreviated. Therefore 12..4(11)..3 equates to 12111111113, specifying a binary number that equals 9559 decimally. * A334836(j) appears in A334771. Table: j i m A334769(i)=n n_16 L k Run Length Recipe ------------------------------------------------------------------------- 2 1 10 151 97 8 *2 12113 3 3 12 543 21f 10 *1 145 4 17 50 10707 29d3 14 2 111231122 5 31 72 33151 817f 16 1 16117 6 63 144 345283 544c3 19 1 11111312242 7 163 308 2213663 21c71f 22 1 1433335 8 351 872 33629695 20125ff 26 2 181212119 9 607 1256 134297599 80137ff 28 1 1.10.1.2.2.1.11 10 1407 2568 1109207903 421d2b5f 31 1 14143112111121115 11 2415 4088 8657682303 20409bf7f 34 1 16161221617 12 5471 9224 73283989519 111010f00f 37 1 13131714484 eof