A389186 - OEIS

%I #27 Feb 26 2026 07:44:38

%S 1,4,11,30,78,137,327,499,1124,1640,3559,5194,10994,16499,34311,53893,

%T 110632,183330,372823,650828,1314966,2398341,4825319,9095375,18252108,

%U 35193132,70517191,137978270,276234842,545423911,1091437319,2166784713,4334814160,8633189254,17268999351

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

%C From _Andrew Howroyd_, Jan 13 2026: (Start)

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

%C 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)

%H Andrew Howroyd, <a href="/A389186/b389186.txt">Table of n, a(n) for n = 0..1000</a>

%H Andrew Howroyd, <a href="/A389186/a389186.txt">Visibility polynomial of the n-sunlet graph</a>, Jan 2026.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SunletGraph.html">Sunlet Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VisibilityPolynomial.html">Visibility Polynomial</a>.

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (3,7,-27,-15,99,-3,-189,60,198,-96,-108,64,24,-16).

%F From _Andrew Howroyd_, Dec 28 2025: (Start)

%F 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).

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

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

%F 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)

%F 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

%t 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 *)

%t 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 *)

%t 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 *)

%o (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

%K nonn,easy

%O 0,2

%A _Eric W. Weisstein_, Sep 25 2025

%E a(16) from _Eric W. Weisstein_, Oct 25 2025

%E a(0)-a(2) prepended and a(17) onward from _Andrew Howroyd_, Dec 28 2025