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%I #20 Sep 08 2022 08:45:51
%S 0,2,10,40,140,462,1470,4580,14080,42922,130130,393120,1184820,
%T 3565382,10717990,32197660,96680360,290215842,870997050,2613690200,
%U 7842468700,23530202302,70596199310,211799782740,635421717840,1906309892762,5719019156770,17157236427280
%N The number of different 4-colorings for the vertices of all triangulated planar polygons on a base with n vertices if the colors of two adjacent boundary vertices are fixed.
%C 1st: The number of different vertex colorings with 4 or 3 colors for n vertices is: (3^(n-1)-2-(-1)^n)/4.
%C 2nd: The number of 3-colorings is: (2^n -3-(-1)^n)/3.
%C The above sequence is the difference between the first and the second one.
%H Vincenzo Librandi, <a href="/A174395/b174395.txt">Table of n, a(n) for n = 3..1000</a>
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,-5,6).
%F a(n) = (3^n - 2^(n+2) + 6 + (-1)^n) / 12.
%F a(n) = 5*a(n-1)-5*a(n-2)-5*a(n-3)+6*a(n-4). G.f.: -2*x^4 / ((x-1)*(x+1)*(2*x-1)*(3*x-1)). - _Colin Barker_, Sep 22 2013
%e n=3 then a(3)=0 as there are no 4-colorings for the only triangle.
%e n=4 then a(4)=2 as there are six good colorings less four 3-colorings for the two triangulated quadrilaterals (4-gons).
%e n=5 then a(5)=10 as there are twenty good colorings less ten 3-colorings for the five triangulated pentagons.
%t CoefficientList[Series[-2 x/((x - 1) (x + 1) (2 x - 1) (3 x - 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 23 2013 *)
%t LinearRecurrence[{5,-5,-5,6},{0,2,10,40},30] (* _Harvey P. Dale_, Aug 29 2015 *)
%o (PARI) Vec(-2*x^4/((x-1)*(x+1)*(2*x-1)*(3*x-1)) + O(x^100)) \\ _Colin Barker_, Sep 22 2013
%o (Magma) [(3^n - 2^(n+2) + 6 + (-1)^n) / 12: n in [3..30]]; // _Vincenzo Librandi_, Sep 23 2013
%Y Equals A081251 (2,6,20...) minus A026644 (2,4,10...)
%K nonn,easy
%O 3,2
%A _Patrick Labarque_, Mar 18 2010, Mar 21 2010
%E More terms from _Colin Barker_, Sep 22 2013
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