A334836 a(n) = A334769(k) where k is the first position of n in A334796.

151, 543, 10707, 33151, 345283, 2213663, 33629695, 134297599, 1109207903, 8657682303, 73283989519

Offset: 2

Author

Michael De Vlieger, May 13 2020

Keywords

• nonn
• more

Comments

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.

Examples

			a(2) = 151; Rotationally symmetrical XOR-triangle generated by 151, replacing 0s with "." for clarity, showing 2 bits to reach the central zero triangle of side length s = 2:
  1 . . 1 . 1 1 1
   1 . 1 1 1 . .
    1 1 . . 1 .
     . 1 . 1 1
      1 1 1 .
       . . 1
        . 1
         1
a(3) = 543; RST generated by 543, showing 3 bits to reach the CZT of side length s = 1 = A334770(3):
  1 . . . . 1 1 1 1 1
   1 . . . 1 . . . .
    1 . . 1 1 . . .
     1 . 1 . 1 . .
      1 1 1 1 1 .
       . . . . 1
        . . . 1
         . . 1
          . 1
           1
		

Links

• Michael De Vlieger, Diagram montage showing XOR-triangles of each a(n) for 2 <= n <= 12.
• Michael De Vlieger, Aspects of XOR-triangles T(a(n))

Crossrefs

Cf. A016777, A070939, A334556, A334769, A334796.