A362030 Irregular triangle read by rows where row n contains the balanced binary words of length 2n interpreted as binary numbers.

1, 2, 3, 5, 6, 9, 10, 12, 7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 135

Offset

1,2

Comments

Within a row, strings are ordered lexicographically, which means the resulting values are ordered numerically.

This is from an idea of David Lovler, which he calls "zigzags". It is a rearrangement of A072601. A072603 lists all the numbers that are not in this sequence. A000984 gives the number of coin flip sequences of length 2,4,6, etc.

Not a permutation of the integers. E.g. 8 never occurs. When there are more 0's than 1's, adding 0's doesn't bring it to balance. - Kevin Ryde, Aug 31 2023

Links

Paolo Xausa, Table of n, a(n) for n = 1..17576 (rows 1..8 of the triangle, flattened).

Example

The first few terms written as binary words with leading 0's: 01, 10, 0011, 0101, 0110, 1001, 1010, 1100, 000111, 001011, 001101, 001110, ... (cf. A368804).
Triangle T(n,k) begins:
   1,  2;
   3,  5,  6,  9, 10, 12;
   7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, ...;
  15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, ...;
  ...

Maple

T:= n-> sort(map(Bits[Join], combinat[permute]([0$n, 1$n])))[]:
seq(T(n), n=1..4);  # Alois P. Heinz, Apr 13 2023

Mathematica

T[n_] := Sort[FromDigits[#, 2] & /@ Permutations[Join[ConstantArray[0, n], ConstantArray[1, n]]]]; Flatten[Table[T[n], {n, 1, 4}]][[1 ;; 64]] (* Robert P. P. McKone, Aug 29 2023 *)

Crossrefs

Columns k=1-2 give: A000225, A083329.

Row sums give A131568.

Main diagonal gives A036563(n+1).

Cf. A000984 (row lengths), A072601, A072603, A368804 (binary).

Sequence in context: A189224 A018762 A059746 * A024620 A232527 A188375

Adjacent sequences: A362027 A362028 A362029 * A362031 A362032 A362033

Keyword

nonn,tabf,easy

Author

Louis Conover, Apr 05 2023

Status

approved