A389186 Number of mutual-visibility sets in the n-sunlet graph.

1, 4, 11, 30, 78, 137, 327, 499, 1124, 1640, 3559, 5194, 10994, 16499, 34311, 53893, 110632, 183330, 372823, 650828, 1314966, 2398341, 4825319, 9095375, 18252108, 35193132, 70517191, 137978270, 276234842, 545423911, 1091437319, 2166784713, 4334814160, 8633189254, 17268999351

Offset

0,2

Comments

From Andrew Howroyd, Jan 13 2026: (Start)

The sequence has been extended to n=0 using the formula.

The visibility polynomial is (1 + x)^n + n*x + n*(3*n-1)*x^2/2 + (4*binomial(n,3) - 3*n*binomial(floor((n-1)/2), 2))*x^3 + n*Sum_{k=0..floor(n/2)-1} ((n-k-2)*x^3 + (floor(n/2)-1-k)^2*x^4)*(1 + x)^k. (End)

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Andrew Howroyd, Visibility polynomial of the n-sunlet graph, Jan 2026.

Eric Weisstein's World of Mathematics, Sunlet Graph.

Eric Weisstein's World of Mathematics, Visibility Polynomial.

Index entries for linear recurrences with constant coefficients, signature (3,7,-27,-15,99,-3,-189,60,198,-96,-108,64,24,-16).

Formula

From Andrew Howroyd, Dec 28 2025: (Start)

G.f.: (1 + x - 8*x^2 - 4*x^3 + 34*x^4 - 49*x^5 - 48*x^6 + 227*x^7 - 34*x^8 - 332*x^9 + 134*x^10 + 164*x^11 - 84*x^12)/((1 - x)^4*(1 + x)^3*(1 - 2*x)*(1 - 2*x^2)^3).

a(2*n) = 4^n + 2*n*(n + 3)*2^n + n*(n^2 - 3*n - 25)/3.

a(2*n + 1) = 2*4^n + (2*n + 1)*(n + 4)*2^n + (2*n + 1)*(n^2 - 5*n - 12)/6.

a(n) = 2^n + n*floor((n+7)\2)*2^floor(n/2) + n*(2*n^2 - 18*n - 137 + (6*n - 63)*(-1)^n)/48. (End)

a(n) = 3*a(n-1)+7*a(n-2)-27*a(n-3)-15*a(n-4)+99*a(n-5)-3*a(n-6)-189*a(n-7)+60*a(n-8)+198*a(n-9)-96*a(n-10)-108*a(n-11)+64*a(n-12)+24*a(n-13)-16*a(n-14). - Eric W. Weisstein, Feb 24 2026

Mathematica

Table[2^n + n Floor[(n + 7)/2] 2^Floor[n/2] + n (2 n^2 - 18 n - 137 + (6 n - 63) (-1)^n)/48, {n, 0, 34}] (* Eric W. Weisstein, Feb 24 2026 *)
LinearRecurrence[{3, 7, -27, -15, 99, -3, -189, 60, 198, -96, -108, 64, 24, -16}, {4, 11, 30, 78, 137, 327, 499, 1124, 1640, 3559, 5194, 10994, 16499, 34311}, {0, 34}] (* Eric W. Weisstein, Feb 24 2026 *)
CoefficientList[Series[(1 + x - 8 x^2 - 4 x^3 + 34 x^4 - 49 x^5 - 48 x^6 + 227 x^7 - 34 x^8 - 332 x^9 + 134 x^10 + 164 x^11 - 84 x^12)/((-1 + x)^4 (-1 + 2 x) (-1 - x + 2 x^2 + 2 x^3)^3), {x, 0, 34}], x] (* Eric W. Weisstein, Feb 24 2026 *)

Prog

(PARI) a(n) = 2^n + n*floor((n+7)\2)*2^floor(n/2) + n*(2*n^2 - 18*n - 137 + (6*n - 63)*(-1)^n)/48; \\ Andrew Howroyd, Dec 28 2025

Crossrefs

Sequence in context: A102281 A269083 A026583 * A339739 A291238 A110034

Adjacent sequences: A389183 A389184 A389185 * A389187 A389188 A389189

Keyword

nonn,easy

Author

Eric W. Weisstein, Sep 25 2025

Extensions

a(16) from Eric W. Weisstein, Oct 25 2025

a(0)-a(2) prepended and a(17) onward from Andrew Howroyd, Dec 28 2025

Status

approved