A335061 - OEIS

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.

%I #18 May 25 2020 23:20:03

%S 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,

%T 151,233,0,40,92,116,138,162,214,254,0,72,140,196,314,370,438,510,543,

%U 599,659,731,805,877,937,993,0,168,854,1022,1379,1483,1589,1693

%N 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.

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

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

%H Michael De Vlieger, <a href="/A335061/b335061.txt">Table of n, a(n) for n = 1..1275</a> (rows 1 <= n <= 24, flattened)

%H A. Barbé, <a href="http://dx.doi.org/10.1016/S0166-218X(00)00211-0">Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2</a>, Discr. Appl. Math. 105(2000), 1-38.

%H Michael De Vlieger, <a href="/A335061/a335061_1.png">Diagram montage</a> showing 486 XOR-triangles T(n,k)>0 for 3 <= n <= 20.

%H Michael De Vlieger, Large 50X25 <a href="/A335061/a335061_2.png">Diagram montage</a> showing 1250 XOR-triangles T(n,k)>0 for 3 <= n <= 24.

%H Rémy Sigrist, <a href="/A335061/a335061.png">Triangles illustrating the initial terms</a>

%H Rémy Sigrist, <a href="/A335061/a335061.gp.txt">PARI program for A335061</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>

%e The first rows are:

%e 0, 1

%e 0

%e 0, 2

%e 0, 6, 11, 13

%e 0, 14

%e 0, 30, 39, 57

%e 0, 8, 54, 62, 83, 91, 101, 109

%e The XOR-triangles corresponding to the 8 terms of row 7 are (with dots instead of 0's for clarity):

%e T(7,1) = 0: T(7,2) = 8: T(7,3) = 54: T(7,4) = 62,

%e . . . . . . . . . . 1 . . . . 1 1 . 1 1 . . 1 1 1 1 1 .

%e . . . . . . . . 1 1 . . 1 . 1 1 . 1 1 . . . . 1

%e . . . . . . 1 . 1 . 1 1 . 1 1 1 . . . 1

%e . . . . 1 1 1 1 . 1 1 . 1 . . 1

%e . . . . . . 1 . 1 1 . 1

%e . . . . 1 1 1 1

%e . . . .

%e T(7,5) = 83: T(7,6) = 91: T(7,7) = 101: T(7,8) = 109:

%e 1 . 1 . . 1 1 1 . 1 1 . 1 1 1 1 . . 1 . 1 1 1 . 1 1 . 1

%e 1 1 1 . 1 . 1 1 . 1 1 . . 1 . 1 1 1 . 1 1 . 1 1

%e . . 1 1 1 . 1 1 . 1 1 1 1 . . 1 . 1 1 .

%e . 1 . . 1 . 1 1 . . 1 . 1 1 . 1

%e 1 1 . 1 1 . . 1 1 . 1 1

%e . 1 . 1 1 . 1 .

%e 1 1 1 1

%t 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 *)

%o (PARI) See Links section.

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

%K nonn,base,tabf

%O 1,5

%A _Rémy Sigrist_, May 21 2020