cp's OEIS Frontend

These are the numbers whose binary representation traces a nonselfcrossing circuit in honeycomb lattice when its bits (from the least to the second most significant bit; the most significant 1-bit is ignored) are interpreted as directions to proceed at each vertex, with an additional condition that the final direction (angle) must be equal to the starting direction of the walk. Because each bit either adds or subtracts 60 degrees from the current phase angle, it implies that for all terms after the initial term a(1)=0 (which stands for an empty path), the difference between the number of 0-bits and 1-bits (when excluding the most significant bit which is always 1) must be either -6 or +6. And indeed, for all n >= 1, A037861(a(n)) is either 5 or -7 as A037861 takes also the most significant bit into account.

Indexing starts from zero, because a(0) = 1 is a special case, indicating an empty path, which thus ends at the same vertex as where it started from.

A258204(n) gives the count of terms with binary width 2n + 1.

Numbers n such that when we start scanning bits in the binary expansion of n, from the most to the least significant end, and when we interpret each bit as to a direction which to turn at each vertex (e.g., 0 = left, 1 = right) when traversing the edges of honeycomb lattice, then, when we have consumed all the bits, we have eventually returned to the same vertex where we started from and none of the other vertices have been visited twice.

Indexing starts from zero, because a(0) = 0 is a special case, indicating an empty path, which certainly ends at the same vertex as where it starts from. For other cases, we always take first a right turn, because the most significant bit is always 1.

The sequence consists of those terms of A255571 whose every A080541/A080542-rotation is also a term of A255571 and in their binary representation the number of 1's is larger than the number of 0's. More precisely, after the initial term a(0)=1 (which stands for an empty path) each term has seven more 1's than 0's in their binary representation, i.e., A037861(a(n)) = -7 for all n >= 1.

Indexing starts from zero, because a(0) = 1 is a special case, indicating an empty path in the honeycomb lattice.

These are capped binary boundary codes for those holeless polyhexes that stay same when they are flipped over and rotated appropriately.

A258205(n) gives the count of terms with binary width 2n + 1.

Numbers n such that when we start scanning bits in the binary expansion of n, from the least to the most significant end, and when we interpret each bit as to a direction which to turn at each vertex (e.g., 0 = left, 1 = right) when traversing the edges of honeycomb lattice, then, when we have consumed all except the most significant 1-bit (which is ignored), we have eventually returned to the same vertex where we started from.

Indexing starts from zero, because a(0) = 1 is a special case, indicating an empty path, which thus ends at the same vertex as where it started from.

If n is a member, then A054429(n) is also a member.

All positive terms belong to A166535, but the reverse is not true (for example, A166535(96) = 136 does not belong to this sequence).

This sequence is infinite as it contains A000975 and A343183.

If m belongs to the sequence, then floor(m/2) also belongs to the sequence.

For any k > 0, the sequence contains A006744(k) positive terms with k binary digits.

This sequence has connections with A258002, A255561 and A255571: these sequences encode in binary nonoverlapping or noncrossing paths in the honeycomb lattice.