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.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

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

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

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

EXAMPLE

The first rows are:

0, 1

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

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 *)

PROG

(PARI) See Links section.

CROSSREFS

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

Sequence in context: A087464 A078048 A362186 * A350462 A357367 A110667

Adjacent sequences: A335058 A335059 A335060 * A335062 A335063 A335064

KEYWORD

nonn,base,tabf

AUTHOR

Rémy Sigrist, May 21 2020

STATUS

approved

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