OFFSET
3,1
COMMENTS
Mutual-visibility sets in the n-antiprism graph have at most 6 vertices.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 3..1000
Andrew Howroyd, Visibility polynomial of the n-antiprism graph, Jan 2026.
Eric Weisstein's World of Mathematics, Antiprism Graph.
Eric Weisstein's World of Mathematics, Visibility Polynomial.
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).
FORMULA
From Andrew Howroyd, Jan 12 2026: (Start)
a(n) = (7*n^6 + 210*n^5 + 1660*n^4 + 3960*n^3 + 11968*n^2 - 24960*n + 5760)/5760 for even n.
a(n) = (7*n^6 + 165*n^5 + 1450*n^4 + 7470*n^3 - 18737*n^2 + 55725*n + 5760)/5760 for odd n >= 5.
G.f.: x^3*(63 + 114*x - 159*x^2 - 168*x^3 + 302*x^4 + 176*x^5 - 387*x^6 - 75*x^7 + 242*x^8 + 19*x^9 - 74*x^10 - 9*x^11 + 11*x^12 + x^13)/((1 - x)^7*(1 + x)^6). (End)
a(n) = a(n-1)+6*a(n-2)-6*a(n-3)-15*a(n-4)+15*a(n-5)+20*a(n-6)-20*a(n-7)-15*a(n-8)+15*a(n-9)+6*a(n-10)-6*a(n-11)-a(n-12)+a(n-13) for n > 16. - Eric W. Weisstein, Feb 16 2026
MATHEMATICA
Join[{63}, Table[1 + (n (30765 - 6769 n + 11430 n^2 + 3110 n^3 + 375 n^4 + 14 n^5 + 15 (-1)^n (-5379 + 2047 n - 234 n^2 + 14 n^3 + 3 n^4)))/11520, {n, 4, 20}]] (* Eric W. Weisstein, Feb 16 2026 *)
Join[{63}, LinearRecurrence[{1, 6, -6, -15, 15, 20, -20, -15, 15, 6, -6, -1, 1}, {177, 396, 912, 1583, 3145, 4759, 8596, 11892, 20113, 26118, 42064, 52131, 80785}, 20]] (* Eric W. Weisstein, Feb 16 2026 *)
CoefficientList[Series[(-63 - 114 x + 159 x^2 + 168 x^3 - 302 x^4 - 176 x^5 + 387 x^6 + 75 x^7 - 242 x^8 - 19 x^9 + 74 x^10 + 9 x^11 - 11 x^12 - x^13)/((-1 + x)^7 (1 + x)^6), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 16 2026 *)
PROG
(PARI) a(n) = if(n==3, -1) + if(n%2, 7*n^6 + 165*n^5 + 1450*n^4 + 7470*n^3 - 18737*n^2 + 55725*n + 5760, 7*n^6 + 210*n^5 + 1660*n^4 + 3960*n^3 + 11968*n^2 - 24960*n + 5760)/5760; \\ Andrew Howroyd, Jan 12 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Sep 25 2025
EXTENSIONS
a(26) from Eric W. Weisstein, Oct 20 2025
a(27) onward from Andrew Howroyd, Jan 12 2026
STATUS
approved