A382534 Number of minimum total dominating sets in the n-flower graph.

3, 9, 9, 36, 15, 81, 21, 36, 27, 225, 33, 36, 39, 441, 45, 36, 51, 729, 57, 36, 63, 1089, 69, 36, 75, 1521, 81, 36, 87, 2025, 93, 36, 99, 2601, 105, 36, 111, 3249, 117, 36

Offset: 1

Author

Eric W. Weisstein, Mar 30 2025

Keywords

• nonn
• easy

Comments

The flower graph is defined for n >= 5. Sequence extended to n = 1 using the formula.

Links

• Eric Weisstein's World of Mathematics, Flower Graph.
• Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Crossrefs

Cf. A017557 (4-section), A016946 (4-section), A017629 (4-section).

Programs

Mathematica

Table[Piecewise[{{36, Mod[n, 4] == 0}, {3 n, Mod[n, 2] == 1}, {9 n^2/4, Mod[n, 4] == 2}}], {n, 20}]
LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {3, 9, 9, 36, 15, 81, 21, 36, 27, 225, 33, 36}, 20]

Formula

a(n) = 3*a(n-4)-3*a(n-8)+a(n-12).

G.f.: -3*x*(1+3*x+3*x^2+12*x^3+2*x^4+18*x^5-2*x^6-24*x^7-3*x^8+3*x^9-x^10+12*x^11)/(x-1)^3/(1+x)^3/(x^2+1)^3 . - R. J. Mathar, Apr 02 2025