A335061 Irregular table read by rows; n-th row corresponds to numbers in the range 0..2^n-1 whose binary expansion (possibly left-padded with 0's up to n binary digits) generates rotationally symmetric XOR-triangles.

0, 1, 0, 0, 2, 0, 6, 11, 13, 0, 14, 0, 30, 39, 57, 0, 8, 54, 62, 83, 91, 101, 109, 0, 126, 151, 233, 0, 40, 92, 116, 138, 162, 214, 254, 0, 72, 140, 196, 314, 370, 438, 510, 543, 599, 659, 731, 805, 877, 937, 993, 0, 168, 854, 1022, 1379, 1483, 1589, 1693

Offset: 1

Author

Rémy Sigrist, May 21 2020

Keywords

• nonn
• base
• tabf

Comments

The n-th row has A060547(n) terms.

Every positive term of A334556, say m, appears in row A070939(m).

Examples

			The first rows are:
    0, 1
    0
    0, 2
    0, 6, 11, 13
    0, 14
    0, 30, 39, 57
    0, 8, 54, 62, 83, 91, 101, 109
The XOR-triangles corresponding to the 8 terms of row 7 are (with dots instead of 0's for clarity):
   T(7,1) = 0:        T(7,2) = 8:        T(7,3) = 54:       T(7,4) = 62,
   . . . . . . .      . . . 1 . . .      . 1 1 . 1 1 .      . 1 1 1 1 1 .
    . . . . . .        . . 1 1 . .        1 . 1 1 . 1        1 . . . . 1
     . . . . .          . 1 . 1 .          1 1 . 1 1          1 . . . 1
      . . . .            1 1 1 1            . 1 1 .            1 . . 1
       . . .              . . .              1 . 1              1 . 1
        . .                . .                1 1                1 1
         .                  .                  .                  .
   T(7,5) = 83:       T(7,6) = 91:       T(7,7) = 101:      T(7,8) = 109:
   1 . 1 . . 1 1      1 . 1 1 . 1 1      1 1 . . 1 . 1      1 1 . 1 1 . 1
    1 1 1 . 1 .        1 1 . 1 1 .        . 1 . 1 1 1        . 1 1 . 1 1
     . . 1 1 1          . 1 1 . 1          1 1 1 . .          1 . 1 1 .
      . 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, Table of n, a(n) for n = 1..1275 (rows 1 <= n <= 24, flattened)
• A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
• Michael De Vlieger, Diagram montage showing 486 XOR-triangles T(n,k)>0 for 3 <= n <= 20.
• Michael De Vlieger, Large 50X25 Diagram montage showing 1250 XOR-triangles T(n,k)>0 for 3 <= n <= 24.
• Rémy Sigrist, Triangles illustrating the initial terms
• Rémy Sigrist, PARI program for A335061
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to cellular automata
• Index entries for sequences related to XOR-triangles

Crossrefs

Cf. A060547 (row length), A070939, A334556.

Programs

Mathematica

Table[Select[Range[0, 2^n - 1], Block[{k = #, w}, (Reverse /@ Transpose[#] /. -1 -> Nothing) == w &@ MapIndexed[PadRight[#, n, -1] &, Set[w, NestList[Map[BitXor @@ # &, Partition[#, 2, 1]] &, PadLeft[IntegerDigits[k, 2], n], n - 1]]]] &], {n, 12}] // Flatten (* Michael De Vlieger, May 24 2020 *)

PARI

See Links section.