A258005 -id:A258005 - cp's OEIS frontend

A258003 Capped binary boundary codes for holeless strictly non-overlapping polyhexes, only the maximal representative from each equivalence class obtained by rotating.

1, 127, 2014, 7918, 31606, 32122, 32188, 126394, 127930, 128476, 486838, 503254, 503482, 505306, 505564, 506332, 511450, 511462, 511708, 511804, 513514, 513772, 513778, 514540, 514804, 514936, 2012890, 2012902, 2013916, 2021098, 2021212, 2022124, 2025196, 2039254, 2043610, 2043622, 2045674, 2045788, 2046700

Offset: 0

Antti Karttunen, May 16 2015

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.

Antti Karttunen, Table of n, a(n) for n = 0..874

Intersection of A257250 and A258002.
Subsequence of A258013.
Subsequence: A258005.
Cf. A037861, A080541, A080542, A255571, A256999, A258204.
Cf. also A258004 (the same terms without the most significant bit, slightly more compact representation).

A258015 Capped binary boundary codes for those fusenes that stay same when flipped over, only the maximal representative from each equivalence class up to rotation.

Original entry on oeis.org

1, 127, 2014, 7918, 31606, 32122, 32188, 126394, 486838, 503482, 505564, 506332, 511708, 511804, 513514, 514936, 2012890, 2021098, 2025196, 2054044, 2055544, 7788250, 8050522, 8051434, 8051548, 8054620, 8075098, 8075110, 8084380, 8104888, 8182636, 8183020, 8185756, 8207218, 8207602, 8214442, 8219596, 8219602, 8231884, 8236516, 8238832

Offset: 0

Antti Karttunen, Jun 01 2015

These are binary boundary codes for fusenes with bilateral symmetry, i.e., those terms k in A258013 for which A256999(A059893(k)) = k. A258018(n) gives the count of terms with binary width 2n + 1.

Differs from its subsequence A258005 for the first time at n=113, as a(113) = 131821024 is the first term not present in A258005.

Antti Karttunen, Table of n, a(n) for n = 0..113

Subsequence of A258013.
Subsequence: A258005.
Cf. A258018.

A258205 Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 1, 8, 5, 20, 11, 61

Offset: 1

Antti Karttunen, May 31 2015

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.

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

(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.

Other identities and observations. For all n >= 1:
a(n) = 2*A258206(n) - A258204(n).
a(n) <= A258018(n).

cp's OEIS Frontend

A258003 Capped binary boundary codes for holeless strictly non-overlapping polyhexes, only the maximal representative from each equivalence class obtained by rotating.

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A258015 Capped binary boundary codes for those fusenes that stay same when flipped over, only the maximal representative from each equivalence class up to rotation.

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A258205 Number of strictly non-overlapping holeless polyhexes of perimeter 2n with bilateral symmetry, counted up to rotation.

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