A303072 Number of minimal total dominating sets in the n-ladder graph.

1, 4, 4, 16, 16, 49, 81, 169, 324, 625, 1296, 2401, 4900, 9409, 18769, 36481, 71824, 141376, 276676, 544644, 1067089, 2099601, 4116841, 8088336, 15880225, 31181056, 61230625, 120209296, 236083225, 463497841, 910168561, 1787091076, 3509140644, 6890328064, 13529411856

Offset: 1

Author

Eric W. Weisstein, Apr 18 2018

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Ladder Graph.
• Eric Weisstein's World of Mathematics, Minimal Total Dominating Set.
• Index entries for linear recurrences with constant coefficients, signature (-1,1,3,7,8,2,6,6,0,0,-6,-6,-2,-8,-7,-3,-1,1,1).

Crossrefs

Row 2 of A303118.

Cf. A290379, A302402, A303054.

Programs

Mathematica

Table[(RootSum[1 - #^2 - #^3 - #^4 + #^6 &, (9 - 18 #^2 + 23 #^3 - 3 #^4 + 32 #^5) #^n &]/229)^2, {n, 40}]
LinearRecurrence[{-1, 1, 3, 7, 8, 2, 6, 6, 0, 0, -6, -6, -2, -8, -7, -3, -1, 1, 1}, {1, 4, 4, 16, 16, 49, 81, 169,324, 625, 1296, 2401, 4900, 9409, 18769, 36481, 71824, 141376, 276676}, 40]
CoefficientList[Series[(-1 - 5 x - 7 x^2 - 13 x^3 - 9 x^4 - x^5 - 4 x^6 + 5 x^7 + 13 x^8 + 14 x^9 + 21 x^10 + 15 x^11 + 12 x^12 + 15 x^13 + 9 x^14 + 3 x^15 - 2 x^17 - x^18)/(-1 - x + x^2 + 3 x^3 + 7 x^4 + 8 x^5 + 2 x^6 + 6 x^7 + 6 x^8 - 6 x^11 - 6 x^12 - 2 x^13 - 8 x^14 - 7 x^15 - 3 x^16 - x^17 + x^18 + x^19), {x, 0, 40}], x]

Formula

a(n) = A253412(n)^2.

G.f.: x*(-1 - 5*x - 7*x^2 - 13*x^3 - 9*x^4 - x^5 - 4*x^6 + 5*x^7 + 13*x^8 + 14*x^9 + 21*x^10 + 15*x^11 + 12*x^12 + 15*x^13 + 9*x^14 + 3*x^15 - 2*x^17 - x^18)/(-1 - x + x^2 + 3*x^3 + 7*x^4 + 8*x^5 + 2*x^6 + 6*x^7 + 6*x^8 - 6*x^11 - 6*x^12 - 2*x^13 - 8*x^14 - 7*x^15 - 3*x^16 - x^17 + x^18 + x^19).