

A174395


The number of different 4colorings for the vertices of all triangulated planar polygons on a base with n vertices if the colors of two adjacent boundary vertices are fixed.


1



0, 2, 10, 40, 140, 462, 1470, 4580, 14080, 42922, 130130, 393120, 1184820, 3565382, 10717990, 32197660, 96680360, 290215842, 870997050, 2613690200, 7842468700, 23530202302, 70596199310, 211799782740, 635421717840, 1906309892762, 5719019156770, 17157236427280
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OFFSET

3,2


COMMENTS

1st: The number of different vertex colorings with 4 or 3 colors for n vertices is: (3^(n1)2(1)^n)/4.
2nd: The number of 3colorings is: (2^n 3(1)^n)/3.
The above sequence is the difference between the first and the second one.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (5,5,5,6).


FORMULA

a(n) = (3^n  2^(n+2) + 6 + (1)^n) / 12.
a(n) = 5*a(n1)5*a(n2)5*a(n3)+6*a(n4). G.f.: 2*x^4 / ((x1)*(x+1)*(2*x1)*(3*x1)).  Colin Barker, Sep 22 2013


EXAMPLE

n=3 then a(3)=0 as there are no 4colorings for the only triangle.
n=4 then a(4)=2 as there are six good colorings less four 3colorings for the two triangulated quadrilaterals (4gons).
n=5 then a(5)=10 as there are twenty good colorings less ten 3colorings for the five triangulated pentagons.


MATHEMATICA

CoefficientList[Series[2 x/((x  1) (x + 1) (2 x  1) (3 x  1)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 23 2013 *)
LinearRecurrence[{5, 5, 5, 6}, {0, 2, 10, 40}, 30] (* Harvey P. Dale, Aug 29 2015 *)


PROG

(PARI) Vec(2*x^4/((x1)*(x+1)*(2*x1)*(3*x1)) + O(x^100)) \\ Colin Barker, Sep 22 2013
(MAGMA) [(3^n  2^(n+2) + 6 + (1)^n) / 12: n in [3..30]]; // Vincenzo Librandi, Sep 23 2013


CROSSREFS

Equals A081251 (2,6,20...) minus A026644 (2,4,10...)
Sequence in context: A244376 A009338 A261473 * A193519 A268329 A223095
Adjacent sequences: A174392 A174393 A174394 * A174396 A174397 A174398


KEYWORD

nonn,easy


AUTHOR

Patrick Labarque, Mar 18 2010, Mar 21 2010


EXTENSIONS

More terms from Colin Barker, Sep 22 2013


STATUS

approved



