A389185 Number of mutual-visibility sets in the n-Moebius ladder.

57, 127, 383, 566, 1527, 1925, 4774, 5330, 12288, 12659, 27405, 26850, 55117, 52265, 102562, 95122, 179608, 163999, 299523, 270414, 479735, 429485, 742686, 660674, 1116784, 988619, 1637457, 1444058, 2348313, 2064849, 3302410, 2897090, 4563640, 3996343, 6208231, 5428966, 8326371, 7273557

Offset

3,1

Comments

Mutual-visibility sets in the n-Moebius ladder have at most 7 vertices. - Andrew Howroyd, Jan 12 2026

Links

Andrew Howroyd, Table of n, a(n) for n = 3..1000

Eric Weisstein's World of Mathematics, Moebius Ladder.

Eric Weisstein's World of Mathematics, Visibility Polynomial.

Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-21,21,35,-35,-35,35,21,-21,-7,7,1,-1).

Formula

From Andrew Howroyd, Jan 12 2026: (Start)

a(n) = (n^7 + 154*n^6 + 2548*n^5 + 47320*n^4 - 58016*n^3 + 9856*n^2 + 2449152*n + 161280)/161280 for even n >= 6.

a(n) = (n^7 + 196*n^6 + 3913*n^5 + 67480*n^4 + 189259*n^3 - 3212636*n^2 + 12144747*n + 161280)/161280 for odd n >= 9.

G.f.: x^3*(57 + 70*x - 143*x^2 - 307*x^3 + 366*x^4 + 587*x^5 - 497*x^6 - 837*x^7 + 231*x^8 + 882*x^9 - 3*x^10 - 471*x^11 + 106*x^12 - 59*x^13 - 167*x^14 + 197*x^15 + 76*x^16 - 84*x^17 - 12*x^18 + 12*x^19)/((1 - x)^8*(1 + x)^7). (End)

Example

From Andrew Howroyd, Jan 12 2026: (Start)
For n >= 14, mutual-visibility sets with 7 vertices are possible. For n = 14, there is only one such set up to isomorphism, illustrated below:
   --x---o---o---o---x---o---o---o---x---o---o---o---x---o- -
     |   |   |   |   |   |   |   |   |   |   |   |   |   | X
   --o---o---x---o---o---o---x---o---o---o---x---o---o---o- -
(End)

Mathematica

Table[Piecewise[{{57, n == 3}, {127, n == 4}, {1527, n == 7}}, 1 + (n (14593899 - 3202780 n + 131243 n^2 + 114800 n^3 + 6461 n^4 + 350 n^5 + 2 n^6 - 21 (-1)^n (461695 - 153452 n + 11775 n^2 + 960 n^3 + 65 n^4 + 2 n^5)))/322560], {n, 3, 40}] (* Eric W. Weisstein, Feb 16 2026 *)
Join[{57, 127, 383, 566, 1527}, LinearRecurrence[{1, 7, -7, -21, 21, 35, -35, -35, 35, 21, -21, -7, 7, 1, -1}, {1925, 4774, 5330, 12288, 12659, 27405, 26850, 55117, 52265, 102562, 95122, 179608, 163999, 299523, 270414}, 20]] (* Eric W. Weisstein, Feb 16 2026 *)
CoefficientList[Series[57 + 127 x + (x^2 (383 + 183 x - 1720 x^2 - 883 x^3 + 4165 x^4 + 1613 x^5 - 6209 x^6 - 1568 x^7 + 5639 x^8 + 999 x^9 - 2960 x^10 - 549 x^11 + 779 x^12 + 267 x^13 - 51 x^14 - 84 x^15 - 12 x^16 + 12 x^17))/((-1 + x)^8 (1 + x)^7), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 16 2026 *)

Prog

(PARI) a(n) = if(n<8, [-63, -8, 0, 0, 12][n-2]) + if(n%2, n^7 + 196*n^6 + 3913*n^5 + 67480*n^4 + 189259*n^3 - 3212636*n^2 + 12144747*n + 161280, n^7 + 154*n^6 + 2548*n^5 + 47320*n^4 - 58016*n^3 + 9856*n^2 + 2449152*n + 161280)/161280; \\ Andrew Howroyd, Jan 12 2026

Crossrefs

Cf. A389184.

Sequence in context: A044625 A304841 A240759 * A154092 A118152 A020186

Adjacent sequences: A389182 A389183 A389184 * A389186 A389187 A389188

Keyword

nonn,easy

Author

Eric W. Weisstein, Sep 25 2025

Extensions

a(26) onward from Andrew Howroyd, Jan 12 2026

Status

approved