A302506 Number of total dominating sets in the n-pan graph.

2, 3, 7, 12, 17, 27, 46, 75, 119, 192, 313, 507, 818, 1323, 2143, 3468, 5609, 9075, 14686, 23763, 38447, 62208, 100657, 162867, 263522, 426387, 689911, 1116300, 1806209, 2922507, 4728718, 7651227, 12379943, 20031168, 32411113, 52442283, 84853394, 137295675

Offset: 1

Author

Eric W. Weisstein, Apr 09 2018

Keywords

• nonn
• easy

Comments

Extended to a(1)-a(2) using the formula/recurrence.

Links

• Eric Weisstein's World of Mathematics, Pan Graph
• Eric Weisstein's World of Mathematics, Total Dominating Set
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Crossrefs

Cf. A000032.

Programs

Mathematica

Table[(3 LucasL[n + 2] + 6 Cos[n Pi/2] - 2 Sin[n Pi/2])/5, {n, 20}]
LinearRecurrence[{1, 0, 1, 1}, {2, 3, 7, 12}, 20]
CoefficientList[Series[(-2 - x - 4 x^2 - 3 x^3)/(-1 + x + x^3 + x^4), {x, 0, 20}], x]

Formula

5*a(n) = 3*A000032(n+2) + 6*cos(n*Pi/2) - 2*sin(n*Pi/2).

a(n) = a(n-1) + a(n-2) + a(n-3) for n > 3.

G.f.: -x*(2 + x + 4*x^2 + 3*x^3)/((1 + x^2)*(x^2 + x - 1)).

E.g.f.: (6*cos(x) - 2*sin(x) - 15 + 3*exp(x/2)*(3*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)))/5. - Stefano Spezia, Jan 03 2023