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A302946
Number of minimal (and minimum) total dominating sets in the 2n-crossed prism graph.
4
4, 36, 196, 1156, 6724, 39204, 228484, 1331716, 7761796, 45239076, 263672644, 1536796804, 8957108164, 52205852196, 304278004996, 1773462177796, 10336495061764, 60245508192804, 351136554095044, 2046573816377476, 11928306344169796, 69523264248641316
OFFSET
1,1
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
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
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]
PROG
(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
CROSSREFS
Cf. A001333, A002203 (sqrt), A090390 (quarter), A287062, A291772, A302941.
Sequence in context: A270084 A272222 A263420 * A220436 A144888 A291643
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Apr 16 2018
EXTENSIONS
a(1) and terms a(6) and beyond from Andrew Howroyd, Apr 16 2018
STATUS
approved