A002943 a(n) = 2*n*(2*n+1).

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, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8190, 8556, 8930

Offset

0,2

Comments

a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and diagonal segments filled in. - Asher Auel, Jan 12 2000

In other words, the edge count of the (n+1) X (n+1) king graph. - Eric W. Weisstein, Jun 20 2017

Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals. (See Example section.)

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

Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].

Leo Tavares, Illustration: Twin Diamond Stars

Eric Weisstein's World of Mathematics, Crown Graph.

Eric Weisstein's World of Mathematics, Edge Count.

Eric Weisstein's World of Mathematics, King Graph.

Eric Weisstein's World of Mathematics, Queen Graph.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

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

a(n) = 2*A014105(n). - 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(8*n+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(2*n) = 2 * A014105(n). - Jon Perry, Oct 27 2014

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 + log(2)/2 - 1. - Amiram Eldar, Feb 22 2022

a(n) = A003154(n+1) - A056220(n+1). - Leo Tavares, Mar 31 2022

Example

64--65--66--67--68--69--70--71--72
|
63  36--37--38--39--40--41--42
|   |                       |
62  35  16--17--18--19--20  43
|   |   |               |   |
61  34  15   4---5---6  21  44
|   |   |    |       |  |   |
60  33  14   3   0   7  22  45
|   |   |    |   |   |  |   |
59  32  13   2---1   8  23  46
|   |   |            |  |   |
58  31  12--11--10---9  24  47
|   |                   |   |
57  30--29--28--27--26--25  48
|                           |
56--55--54--53--52--51--50--49

Maple

A002943 := proc(n)
    2*n*(2*n+1) ;
end proc: # R. J. Mathar, Jun 28 2013

Mathematica

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

Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.

Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.

Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.

Cf. A003154, A056220.

Sequence in context: A338120 A077539 A068377 * A009946 A290154 A094274

Adjacent sequences: A002940 A002941 A002942 * A002944 A002945 A002946

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

Formula fixed by Reinhard Zumkeller, Apr 09 2010

Status

approved