A258205 - OEIS

A258205

Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

0, 0, 1, 0, 1, 1, 3, 1, 8, 5, 20, 11, 61, 35, 167, 97, 511, 297, 1503

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OFFSET

1,7

COMMENTS

This sequence counts by perimeter length those holeless polyhexes that stay same when they are flipped over and rotated appropriately.

For n >= 1, a(n) gives the total number of terms k in A258005 with binary width = 2n + 1, or equally, with A000523(k) = 2n.

LINKS

Table of n, a(n) for n=1..19.

Tyler Satchel Orden, C++ program computing a(1)-a(19) (also A258204 and A258206)

FORMULA

Other identities and observations. For all n >= 1:

a(n) = 2*A258206(n) - A258204(n).

a(n) <= A258018(n).

PROG

(Scheme)

(define (A258205 n) (let loop ((k (+ 1 (expt 2 (+ n n)))) (c 0)) (cond ((pow2? k) c) (else (loop (+ 1 k) (+ c (if (isA258005? k) 1 0)))))))

(define (pow2? n) (let loop ((n n) (i 0)) (cond ((zero? n) #f) ((odd? n) (and (= 1 n) i)) (else (loop (/ n 2) (1+ i)))))) ;; Gives non-false only when n is a power of two.

;; Code for isA258005? given in A258005.

(C++) // a(14)-a(19) were computed by an exhaustive depth-first enumeration of the 2n-step closed self-avoiding walks on the honeycomb lattice, with polygons collected into equivalence classes under the 12-element point group and each class tested for reflection symmetry. See the a-file link for the program. - Tyler Satchel Orden, Jul 13 2026

CROSSREFS

Cf. A057779, A258005, A258018, A258204, A258206.

Sequence in context: A172157 A209136 A206800 * A258018 A188939 A062196

Adjacent sequences: A258202 A258203 A258204 * A258206 A258207 A258208

KEYWORD

nonn,more

AUTHOR

Antti Karttunen, May 31 2015

EXTENSIONS

a(14)-a(19) from Tyler Satchel Orden, Jul 13 2026

STATUS

approved