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

Offset: 1

Author

Antti Karttunen, May 31 2015

Keywords

• nonn
• more

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.

Crossrefs

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

Programs

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.

Formula

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

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

a(n) <= A258018(n).