A302761 Number of total dominating sets in the n-barbell graph.

1, 4, 23, 136, 707, 3312, 14527, 61264, 252515, 1027192, 4147343, 16674984, 66887875, 267960544, 1072726271, 4292804896, 17175281987, 68709777768, 274857460111, 1099468636600, 4397956334051, 17591997301264, 70368349913663, 281474154627696, 1125898195567267

Offset: 1

Author

Eric W. Weisstein, Apr 12 2018

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Barbell Graph
• Eric Weisstein's World of Mathematics, Total Dominating Set
• Index entries for linear recurrences with constant coefficients, signature (11,-47,101,-116,68,-16).

Crossrefs

Cf. A060867, A302650.

Programs

Magma

[(2^n-n)^2 - (2^n-2*n): n in [1..30]]; // Vincenzo Librandi, Apr 15 2018

Mathematica

Array[(2^# - #)^2 - (2^# - 2 #) &, 30] (* Michael De Vlieger, Apr 14 2018 *)
Table[(2^n - n)^2 - (2^n - 2*n), {n, 30}]
LinearRecurrence[{11, -47, 101, -116, 68, -16}, {1, 4, 23, 136, 707, 3312}, 30]
CoefficientList[Series[(1 - 7 x + 26 x^2 - 30 x^3 + 4 x^4)/((-1 + x)^3 (-1 + 2 x)^2 (-1 + 4 x)), {x, 0, 30}], x] (* Eric W. Weisstein, Apr 16 2018 *)

PARI

a(n)={(2^n-n)^2 - (2^n-2*n)} \\ Andrew Howroyd, Apr 14 2018

PARI

Vec(x*(1 - 7*x + 26*x^2 - 30*x^3 + 4*x^4) / ((1 - x)^3*(1 - 2*x)^2*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Apr 15 2018

Formula

a(n) = (2^n-n)^2 - (2^n-2*n). - Andrew Howroyd, Apr 14 2018

From Colin Barker, Apr 15 2018: (Start)

G.f.: x*(1 - 7*x + 26*x^2 - 30*x^3 + 4*x^4) / ((1 - x)^3*(1 - 2*x)^2*(1 - 4*x)).

a(n) = 11*a(n-1) - 47*a(n-2) + 101*a(n-3) - 116*a(n-4) + 68*a(n-5) - 16*a(n-6) for n>6. (End)

E.g.f.: exp(x)*(exp(3*x) + x*(3 + x) - exp(x)*(1 + 4*x)). - Stefano Spezia, Sep 06 2023

Extensions

a(1)-a(2) and a(11)-a(25) from Andrew Howroyd, Apr 14 2018