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A002943 2*n*(2*n+1). 44
0, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and diagonal segments filled in. - Asher Auel (asher.auel(AT)reed.edu), Jan 12 2000

Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals.

The identity (4*n+1)^2-(4*n^2+2*n)*(2)^2 = 1 can be written as A016813(n)^2-a(n)*2^2 = 1. - Vincenzo Librandi, Jul 20 2010 - Nov 25 2012

Starting with "6" = binomial transform of [6, 14, 8, 0, 0, 0,...]. - Gary W. Adamson, Aug 27 2010

The hyper-Wiener index of the crown graph G(n) (n>=3). The crown graph G(n) is the graph with vertex set {x(1), x(2), ..., x(n), y(1), y(2), ..., y(n)} and edge set {(x(i), y(j)): 1<=i,j<=n, i =/ j} (= the complete bipartite graph K(n,n) with horizontal edges removed). The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - Emeric Deutsch, Aug 29 2013

Sum of the numbers from n to 3n. - Wesley Ivan Hurt, Oct 27 2014

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Queen's Tour Graph

Eric Weisstein's World of Mathematics,Crown Graph.

Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 4*n^2 + 2*n.

a(n) = A014105(n)*2. - Omar E. Pol, May 21 2008

a(n) = floor((2*n + 1/2)^2). - Reinhard Zumkeller, Feb 20 2010

a(n) = A007494(n) + A173511(n) = A007742(n) + n. - Reinhard Zumkeller, Feb 20 2010

a(n) = 8*n+a(n-1)-2 with a(0)=0. - Vincenzo Librandi, Jul 20 2010

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Aug 11 2011

a(n+1) = A045896(2*n+1). - Reinhard Zumkeller, Dec 12 2011

G.f.: 2*x*(3+x)/(1-x)^3. - Colin Barker, Jan 14 2012

Sum_{n>=1} 1/a(n) = 1-log(2). Sum_{n>=1} 1/a(n)^2 = 2*log(2)+Pi^2/6-3. - R. J. Mathar, Jan 15 2013

a(n) = A118729(8n+5). - Philippe Deléham, Mar 26 2013

a(n) = 1*A001477(n) + 2*A000217(n) + 3*A000290(n). - J. M. Bergot, Apr 23 2014

a(n) = 2 * A000217(2n) = 2 * A014105(n). - Jon Perry, Oct 27 2014

EXAMPLE

16 17 18 19 ...

15 4 5 6 ...

14 3 0 7 ...

13 2 1 8 ...

MAPLE

A002943 := proc(n)

    2*n*(2*n+1) ;

end proc: # R. J. Mathar, Jun 28 2013

MATHEMATICA

s=0; lst={s}; Do[s+=n++ +6; AppendTo[lst, s], {n, 0, 7!, 8}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)

LinearRecurrence[{3, -3, 1}, {0, 6, 20}, 40] (* or *) Table[2n(2n+1), {n, 0, 40}] (* Harvey P. Dale, Aug 11 2011 *)

PROG

(PARI) a(n)=2*n*(2*n+1) \\ Charles R Greathouse IV, Nov 20 2012

(MAGMA) [ 4*n^2+2*n: n in [0..50]]; // Vincenzo Librandi, Nov 25 2012

(Haskell)

a002943 n = 2 * n * (2 * n + 1)  -- Reinhard Zumkeller, Jan 12 2014

CROSSREFS

Cf. A007742, A033954, A046092, A054000, A014105, A007395, A016813.

Same as A033951 except start at 0.

Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.

Sequence in context: A097811 A143711 A077539 * A068377 A009946 A094274

Adjacent sequences:  A002940 A002941 A002942 * A002944 A002945 A002946

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Formula fixed by Reinhard Zumkeller, Apr 09 2010

STATUS

approved

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Last modified October 30 11:52 EDT 2014. Contains 248800 sequences.