cp's OEIS Frontend

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.

Differs from A258002 for the first time at n=6622, where a(6622) = 69131119 which is missing from A258002 because that number codes for one of the 26 different orientations of the same 26-edge six-hex polyhex where the two hexes at the ends of the pattern touch each other. This pattern is isomorphic to benzenoid [6]Helicene (up to chirality, see the illustrations at Wikipedia-page).

The terms in this sequence are those whose binary representation can be rewritten to 127 (in binary "1111111", which encodes the boundary of a single hexagon) with an appropriate sequence of invocations of recurrences A254109 and A258009. However, there are some intricacies as how this should be done to get correct results. (Please see Kovič paper.)

Note that the papers in literature employ different, "Boundary Edges Code for Benzenoid Systems" (BEC for short) but to which these binary boundary codes can be directly related via their run-lengths.

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.

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.