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Number of connected dominating sets in the 2n-crossed prism graph.
2

%I #14 Feb 16 2025 08:33:46

%S 115,1063,9121,75607,611569,4857223,38034241,294475447,2258978449,

%T 17196401383,130059675361,978211787287,7322040929329,54576195433543,

%U 405286730532481,2999780651211127,22137879320864209,162941058582753703,1196418733436205601

%N Number of connected dominating sets in the 2n-crossed prism graph.

%H Andrew Howroyd, <a href="/A287430/b287430.txt">Table of n, a(n) for n = 2..200</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConnectedDominatingSet.html">Connected Dominating Set</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrossedPrismGraph.html">Crossed Prism Graph</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14, -49).

%F From _Andrew Howroyd_, Sep 05 2017: (Start)

%F a(n) = 7^n + 240*n*7^(n-3) for n > 2.

%F a(n) = 14*a(n-1) - 49*a(n-2) for n > 4.

%F G.f.: x^2*(115 - 547*x - 126*x^2)/(1 - 7*x)^2.

%F (End)

%t Join[{115}, Table[7^(n - 3) (343 + 240 n), {n, 3, 20}]]

%t LinearRecurrence[{14, -49}, {115, 1063, 9121}, 19] (* amended by _Georg Fischer_, Apr 03 2019 *)

%t CoefficientList[Series[(115 - 547 x - 126 x^2)/(-1 + 7 x)^2, {x, 0, 20}], x]

%o (PARI) Vec((115 - 547*x - 126*x^2)/(1 - 7*x)^2 + O(x^20)) \\ _Andrew Howroyd_, Sep 05 2017

%Y Cf. A287062, A290757.

%K nonn,changed

%O 2,1

%A _Eric W. Weisstein_, May 25 2017

%E Terms a(6) and beyond from _Andrew Howroyd_, Sep 05 2017