OFFSET
1,1
COMMENTS
For n >= 4, the visibility polynomial is (1 + x)^(2*n) + 2*(n + 1)*x^n - ((n^2 + n + 2) + (2*n + 4)*x + 2*x^2)*(1 + x)^(n-2)*x^n + (n*(n-1)*(n-2)*(n-3)/4 + n*(n-1)*(n-2)*x + n*(2*n - 3)*x^2 + 2*n*x^3 + x^4)*x^(2*n-4). - Andrew Howroyd, Jan 12 2026
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
Andrew Howroyd, Visibility polynomial of the n-crown graph, Jan 2026.
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Visibility Polynomial.
Index entries for linear recurrences with constant coefficients, signature (15,-96,346,-777,1131,-1070,636,-216,32).
FORMULA
From Andrew Howroyd, Jan 11 2026: (Start)
a(n) = 4^n - (n^2 + 3*n + 8)*2^(n-2) + (n^4 - 2*n^3 + 7*n^2 + 6*n + 12)/4 for n >= 4.
G.f.: x*(3 - 38*x + 219*x^2 - 725*x^3 + 1438*x^4 - 1459*x^5 - 222*x^6 + 2822*x^7 - 4028*x^8 + 2984*x^9 - 1168*x^10 + 192*x^11)/((1 - x)^5*(1 - 2*x)^3*(1 - 4*x)). (End)
MATHEMATICA
Table[Piecewise[{{3, n == 1}, {7, n == 2}, {36, n == 3}}, (12 + 4^(n + 1) + 6 n + 7 n^2 - 2 n^3 + n^4 - 2^n (8 + 3 n + n^2))/4], {n, 20}] (* Eric W. Weisstein, Feb 16 2026 *)
Join[{3, 7, 36}, Drop[LinearRecurrence[{15, -96, 346, -777, 1131, -1070, 636, -216, 32}, {4, 11, 42, 181, 788, 3395, 14416, 60287, 248730}, 20], 3]] (* Eric W. Weisstein, Feb 16 2026 *)
CoefficientList[Series[(-3 + 38 x - 219 x^2 + 725 x^3 - 1438 x^4 + 1459 x^5 + 222 x^6 - 2822 x^7 + 4028 x^8 - 2984 x^9 + 1168 x^10 - 192 x^11)/((-1 + x)^5 (-1 + 2 x)^3 (-1 + 4 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Feb 16 2026 *)
PROG
(PARI) a(n) = if(n<4, [3, 7, 36][n], 4^n - (n^2 + 3*n + 8)*2^(n-2) + (n^4 - 2*n^3 + 7*n^2 + 6*n + 12)/4); \\ Andrew Howroyd, Jan 11 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Dec 14 2025
EXTENSIONS
a(13) from Eric W. Weisstein, Dec 31 2025
a(14) onward from Andrew Howroyd, Jan 11 2026
STATUS
approved