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 A002943 a(n) = 2*n*(2*n+1). 67
 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 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 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 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. Sequence in context: A097811 A143711 A077539 * A068377 A009946 A290154 Adjacent sequences:  A002940 A002941 A002942 * A002944 A002945 A002946 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Formula fixed by Reinhard Zumkeller, Apr 09 2010 STATUS approved

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Last modified October 18 01:01 EDT 2018. Contains 316297 sequences. (Running on oeis4.)