A302946 Number of minimal (and minimum) total dominating sets in the 2n-crossed prism graph.

4, 36, 196, 1156, 6724, 39204, 228484, 1331716, 7761796, 45239076, 263672644, 1536796804, 8957108164, 52205852196, 304278004996, 1773462177796, 10336495061764, 60245508192804, 351136554095044, 2046573816377476, 11928306344169796, 69523264248641316

Offset: 1

Author

Eric W. Weisstein, Apr 16 2018

Keywords

• nonn
• easy

Comments

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

Since minimal and minimum total dominating sets are equivalent, the crossed prism graphs could be said to be "well totally dominated".

Links

• Eric Weisstein's World of Mathematics, Crossed Prism Graph
• Eric Weisstein's World of Mathematics, Total Dominating Set
• Eric Weisstein's World of Mathematics, Well-Covered Graph
• Index entries for linear recurrences with constant coefficients, signature (5,5,-1).

Crossrefs

Cf. A001333, A002203 (sqrt), A090390 (quarter), A287062, A291772, A302941.

Programs

Mathematica

Table[2 (ChebyshevT[n, 3] + (-1)^n), {n, 20}]
Table[4 (-1)^n ChebyshevT[n, I]^2, {n, 20}]
LinearRecurrence[{5, 5, -1}, {4, 36, 196}, 20]
CoefficientList[Series[-4 (-1 - 4 x + x^2)/(1 - 5 x - 5 x^2 + x^3), {x, 0, 20}], x]

PARI

Vec(4*(1 + 4*x - x^2)/((1 + x)*(1 - 6*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Apr 16 2018

PARI

a(n) = 2*(polchebyshev(n,1,3) + (-1)^n); \\ Michel Marcus, Apr 17 2018

Formula

From Andrew Howroyd, Apr 16 2018: (Start)

a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3).

G.f.: 4*x*(1 + 4*x - x^2)/((1 + x)*(1 - 6*x + x^2)).

a(n) = 4*A090390(n) = 4*A001333(n)^2. (End)

a(n) = 2*(chebyshevT(n,3) + (-1)^n). - Eric W. Weisstein, Apr 17 2018

a(n) = 4*(-1)^n*chebyshevT(n,i)^2, where i is the imaginary unit. - Eric W. Weisstein, Apr 17 2018

E.g.f.: 2*(exp(-x) + exp(3*x)*cosh(2*sqrt(2)*x) - 2). - Stefano Spezia, Aug 03 2024

Extensions

a(1) and terms a(6) and beyond from Andrew Howroyd, Apr 16 2018