login
This site is supported by donations to The OEIS Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A179824 Chromatic polynomial of the star graph on 4 vertices (claw graph) and the path graph on 4 vertices. 3
2, 24, 108, 320, 750, 1512, 2744, 4608, 7290, 11000, 15972, 22464, 30758, 41160, 54000, 69632, 88434, 110808, 137180, 168000, 203742, 244904, 292008, 345600, 406250, 474552, 551124, 636608, 731670, 837000, 953312, 1081344, 1221858 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

To generate a Pythagorean triangle one uses (a,b) to get sides b^2-a^2, 2*a*b, and a^2+b^2, having a perimeter of 2*b*(a+b). If for a one uses the triangular number n*(n+1)/2 and for b the next triangular number (n+1)*(n+2)/2, the perimeter of the triangle so formed is (n+1)^3 * (n+2), which will give the same results as this sequence starting at the second term. - J. M. Bergot, Apr 01 2012

Define b(0)=0 and b(n)=A179824(n+1) for n > 0.  Then b(n) is the number of 4-tuples (w,x,y,z) having all terms in {0,...,n} and no two consecutive terms equal. - Clark Kimberling, May 31 2012

Let n points in the plane each become the centers of n-1 concentric circles, circles that pass through only one of each of the other points. The maximum number of intersections of these circles is this sequence. [The solution was given by Andrew Weimholt in the Sequence Fans Mailing List] - J. M. Bergot, Mar 10 2014

Both the 'claw graph', a graph with 4 vertices where one vertex is adjacent to the other three, and the path graph on 4 vertices (per Clark Kimberling's comment), have this sequence as their chromatic polynomial, or the number of proper colorings of the graph using at most n colors. This is the standard example of two graphs which are not isomorphic, but which have the same chromatic polynomial.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 2..1000

Andrew Weimholt, Re: Intersecting circles, SeqFan post Mar 09 2014.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

a(n) = n*(n-1)^3. - Jaime Soffer (jaime.soffer(AT)gmail.com), Jul 30 2010

G.f.: 2*x^2*(1 + 7*x + 4*x^2)/(1-x)^5. - Colin Barker, Jan 30 2012

a(n) = 2*A019582(n). - R. J. Mathar, Jun 09 2013

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Mar 12 2014

EXAMPLE

From Jack W Grahl, Jul 16 2018: (Start)

Consider the claw graph, which has vertices A, B, C, D, and edges AB, AC, AD. To color this graph with 3 colors, we can choose any of the 3 colors for A. Then each of the other vertices can be colored with any of the remaining two colors, giving 3 * 2 * 2 * 2 = 24 choices in all.

Similarly, consider the path graph with the same vertices and edges AB, BC, CD. We have 3 choices for the color of A, then 2 choices for the color of B (any color except that chosen for A), 2 choices for the color of C (any color except B's) etc. (End)

MATHEMATICA

CoefficientList[Series[2 (1 + 7 x + 4 x^2)/(1 - x)^5, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 12 2014 *)

Table[n^3+n^4, {n, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {2, 24, 108, 320, 750}, 40] (* Harvey P. Dale, Sep 05 2015 *)

PROG

(Haskell) let f n = [ (x, a, b, c) | let t = [1..n], x <- t, a <- t, x /= a, b <- t, x /= b, c <- t, x /= c ] in map (length.f) [2..]

(Haskell) let f n = n*(n-1)^3 in map f [2..]

(PARI) a(n) = n*(n-1)^3 \\ Charles R Greathouse IV, Mar 11 2014

(MAGMA) I:=[2, 24, 108, 320, 750]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Mar 12 2014

CROSSREFS

Sequence in context: A121199 A009538 A009556 * A034310 A060817 A045820

Adjacent sequences:  A179821 A179822 A179823 * A179825 A179826 A179827

KEYWORD

nonn,easy

AUTHOR

Jaime Soffer (jaime.soffer(AT)gmail.com), Jul 28 2010

EXTENSIONS

Name edited by Jack W Grahl, Jul 16 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)