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Number of independent sets of nodes in graph K_6 X C_n (n > 2).
2

%I #28 Sep 13 2023 12:23:07

%S 7,1,43,229,1447,8881,54763,337429,2079367,12813601,78961003,

%T 486579589,2998438567,18477210961,113861704363,701647437109,

%U 4323746327047,26644125399361,164188498723243,1011775117738789,6234839205156007,38420810348674801,236759701297204843

%N Number of independent sets of nodes in graph K_6 X C_n (n > 2).

%H Vincenzo Librandi, <a href="/A051931/b051931.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,7,1).

%F a(n) = 5*a(n-1) + 7*a(n-2) + a(n-3).

%F G.f.: (7 - 34*x - 11*x^2) / ((1 + x)*(1 - 6*x - x^2)). - _Colin Barker_, Apr 18 2012

%F From _Colin Barker_, Nov 24 2017: (Start)

%F a(n) = (3 - sqrt(10))^n + (3 + sqrt(10))^n + 5 for n even.

%F a(n) = (3 - sqrt(10))^n + (3 + sqrt(10))^n - 5 for n odd.

%F (End)

%t CoefficientList[Series[(7-34*x-11*x^2)/((1+x)*(1-6*x-x^2)),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 27 2012 *)

%o (Magma) I:=[7, 1, 43]; [n le 3 select I[n] else 5*Self(n-1)+7*Self(n-2)+Self(n-3): n in [1..25]]; // _Vincenzo Librandi_, Apr 27 2012

%o (PARI) Vec((7 - 34*x - 11*x^2) / ((1 + x)*(1 - 6*x - x^2)) + O(x^40)) \\ _Colin Barker_, Nov 24 2017

%Y Row 6 of A287376.

%K easy,nonn

%O 0,1

%A _Stephen G Penrice_, Dec 19 1999

%E More terms from _James A. Sellers_, Dec 20 1999