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Number of independent sets of vertices in graph K_5 X C_n (n > 2).
2

%I #27 Sep 13 2023 12:22:39

%S 6,1,31,136,731,3771,19606,101781,528531,2744416,14250631,73997551,

%T 384238406,1995189561,10360186231,53796120696,279340789731,

%U 1450500069331,7531841136406,39109705751341,203080369893131,1054511555216976,5475638145978031,28432702285107111

%N Number of independent sets of vertices in graph K_5 X C_n (n > 2).

%H Vincenzo Librandi, <a href="/A051930/b051930.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 (4,6,1).

%F a(n) = 4*a(n-1) + 6*a(n-2) + a(n-3).

%F G.f.: (6 - 23*x - 9*x^2) / ((1 + x)*(1 - 5*x - x^2)). - _Colin Barker_, May 22 2012

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

%F a(n) = ((5-sqrt(29))/2)^n + ((5+sqrt(29))/2)^n + 4 for n even.

%F a(n) = ((5-sqrt(29))/2)^n + ((5+sqrt(29))/2)^n - 4 for n odd.

%F (End)

%t LinearRecurrence[{4,6,1},{6,1,31},30] (* _Vincenzo Librandi_, Jun 17 2012 *)

%o (Magma) I:=[6, 1, 31]; [n le 3 select I[n] else 4*Self(n-1)+6*Self(n-2)+Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jun 17 2012

%o (PARI) Vec((6 - 23*x - 9*x^2) / ((1 + x)*(1 - 5*x - x^2)) + O(x^40)) \\ _Colin Barker_, Nov 24 2017

%Y Row 5 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