A302654 Number of minimum total dominating sets in the n-path graph.

0, 1, 2, 1, 1, 4, 3, 1, 2, 9, 4, 1, 3, 16, 5, 1, 4, 25, 6, 1, 5, 36, 7, 1, 6, 49, 8, 1, 7, 64, 9, 1, 8, 81, 10, 1, 9, 100, 11, 1, 10, 121, 12, 1, 11, 144, 13, 1, 12, 169, 14, 1, 13, 196, 15, 1, 14, 225, 16, 1, 15, 256, 17, 1, 16, 289, 18, 1, 17, 324, 19, 1, 18, 361, 20, 1

Offset: 1

Author

Eric W. Weisstein, Apr 11 2018

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
• Eric Weisstein's World of Mathematics, Path Graph.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Crossrefs

Row 1 of A303293.

Cf. A302653, A302655.

Programs

Mathematica

Table[Piecewise[{{1, Mod[n, 4] == 0}, {((n + 2)/4)^2, Mod[n, 4] == 2}, {(n - 1)/4, Mod[n, 4] == 1}, {(n + 5)/4, Mod[n, 4] == 3}}], {n, 20}]
Table[((-1)^n (n - 2)^2 + (6 + n)^2 - 2 (n - 2) (n + 6) Cos[n Pi/2] - 48 Sin[n Pi/2])/64, {n, 20}]
LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 2, 1, 1, 4, 3, 1, 2, 9, 4, 1}, 20]

PARI

concat(0, Vec(x^2*(1 + 2*x + x^2 + x^3 + x^4 - 3*x^5 - 2*x^6 - x^7 + x^9 + x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3) + O(x^70))) \\ Colin Barker, Dec 25 2019

Formula

a(n) = ((-1)^n*(n - 2)^2 + (6 + n)^2 - 2*(n - 2)*(n + 6)*cos(n*Pi/2) - 48*sin(n*Pi/2))/6.

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

G.f.: x^2*(1 + 2*x + x^2 + x^3 + x^4 - 3*x^5 - 2*x^6 - x^7 + x^9 + x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3). - Colin Barker, Dec 25 2019

E.g.f.: ((12 + x^2)*cos(x) + (20 + 8*x + x^2)*cosh(x) + (5*x - 24)*sin(x) + (16 + 5*x)*sinh(x) - 32)/32. - Stefano Spezia, Jun 10 2025