A347650 Number of minimum total dominating sets in the n-pan graph (for n > 2).

1, 2, 3, 2, 3, 8, 5, 2, 5, 18, 7, 2, 7, 32, 9, 2, 9, 50, 11, 2, 11, 72, 13, 2, 13, 98, 15, 2, 15, 128, 17, 2, 17, 162, 19, 2, 19, 200, 21, 2, 21, 242, 23, 2, 23, 288, 25, 2, 25, 338, 27, 2, 27, 392, 29, 2, 29, 450, 31, 2, 31, 512, 33, 2, 33, 578, 35, 2, 35, 648

Offset: 1

Author

Eric W. Weisstein, Sep 09 2021

Keywords

• nonn

Comments

Sequence extended to a(1) using the formula/recurrence.

The total domination number is given by A004524(n + 2). - Andrew Howroyd, Jun 11 2025

Links

• Eric Weisstein's World of Mathematics, Pan 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. A004524, A302506, A303148.

Programs

Mathematica

Table[Piecewise[{{(n + 3)/2, Mod[n, 4] == 3}, {2, Mod[n, 4] == 0}, {(n + 1)/2, Mod[n, 4] == 1}, {(n + 2)^2/8, Mod[n, 4] == 2}}], {n, 20}]
LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {1, 2, 3, 2, 3, 8, 5, 2, 5, 18, 7, 2}, 20]
CoefficientList[Series[(-1 - 2 x - 3 x^2 - 2 x^3 - 2 x^5 + 4 x^6 + 4 x^7 + x^8 - x^10 - 2 x^11)/(-1 + x^4)^3, {x, 0, 20}], x]

Formula

a(n) = (n+3)/2 for n = 3 (mod 4)

= 2 for n = 0 (mod 4)

= (n+1)/2 for n = 1 (mod 3)

= (n+2)^2/8 for n = 2 (mod 4).

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

G.f.: x*(-1-2*x-3*x^2-2*x^3-2*x^5+4*x^6+4*x^7+x^8-x^10-2*x^11)/(-1+x^4)^3.