OFFSET
1,1
COMMENTS
A subset of the vertex set of a complete tripartite graph is not a mutual-visibility set if it contains all the vertices of two parts and at least two vertices of the third part. - Christian Sievers, Oct 21 2025
LINKS
Christian Sievers, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Complete Tripartite Graph.
Eric Weisstein's World of Mathematics, Visibility Polynomial.
Index entries for linear recurrences with constant coefficients, signature (12,-37,42,-16).
FORMULA
a(n) = 8^n - 3*2^n + 3*n + 5 for n>=2. - Christian Sievers, Oct 21 2025
a(n) = 12*a(n-1)-37*a(n-2)+42*a(n-3)-16*a(n-4) for n >= 6. - Eric W. Weisstein, Oct 23 2025
G.f.: x*(8 - 33*x + 42*x^2 + 36*x^3 - 32*x^4)/((1 - x)^2*(1 - 2*x)*(1 - 8*x)). - Eric W. Weisstein, Oct 23 2025 [corrected by Stefano Spezia, Dec 01 2025]
E.g.f.: exp(x)*(5 - 3*exp(x) + exp(7*x) + 3*x) - 2*x - 3. - Stefano Spezia, Dec 01 2025
MATHEMATICA
Table[Piecewise[{{8, n == 1}}, 5 - 3 2^n + 8^n + 3 n], {n, 20}] (* Eric W. Weisstein, Oct 23 2025 *)
ReplacePart[LinearRecurrence[{12, -37, 42, -16}, {10, 63, 502, 4065}, 20], 1 -> 8] (* Eric W. Weisstein, Oct 23 2025 *)
CoefficientList[Series[-2 + 8/(1 - 8 x) + 3/(-1 + x)^2 - 5/(-1 + x) + 6/(-1 + 2 x), {x, 0, 20}], x] (* Eric W. Weisstein, Oct 23 2025 *)
PROG
(PARI) a(n)=8^n-(n>1)*(3*2^n-3*n-5) \\ Christian Sievers, Oct 21 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Oct 13 2025
EXTENSIONS
a(9) onwards from Christian Sievers, Oct 21 2025
STATUS
approved