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

1,2

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).

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

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 *)

Prog

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

Crossrefs

Cf. A060867, A302650.

Sequence in context: A239399 A193808 A038723 * A091640 A237362 A067110

Adjacent sequences: A302758 A302759 A302760 * A302762 A302763 A302764

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 12 2018

Extensions

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

Status

approved