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A174395 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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

1st: The number of different vertex colorings with 4 or 3 colors for n vertices is: (3^(n-1)-2-(-1)^n)/4.

2nd: The number of 3-colorings 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(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

EXAMPLE

n=3 then a(3)=0 as there are no 4-colorings for the only triangle.

n=4 then a(4)=2 as there are six good colorings less four 3-colorings for the two triangulated quadrilaterals (4-gons).

n=5 then a(5)=10 as there are twenty good colorings less ten 3-colorings 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/((x-1)*(x+1)*(2*x-1)*(3*x-1)) + 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 * A320526 A193519 A268329

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

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Last modified December 7 04:29 EST 2022. Contains 358649 sequences. (Running on oeis4.)