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%I #117 Oct 27 2023 22:00:43
%S 0,6,20,42,72,110,156,210,272,342,420,506,600,702,812,930,1056,1190,
%T 1332,1482,1640,1806,1980,2162,2352,2550,2756,2970,3192,3422,3660,
%U 3906,4160,4422,4692,4970,5256,5550,5852,6162,6480,6806,7140,7482,7832,8190,8556,8930
%N a(n) = 2*n*(2*n+1).
%C 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
%C In other words, the edge count of the (n+1) X (n+1) king graph. - _Eric W. Weisstein_, Jun 20 2017
%C Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals. (See Example section.)
%C 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
%C Starting with "6" = binomial transform of [6, 14, 8, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 27 2010
%C 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
%C Sum of the numbers from n to 3n. - _Wesley Ivan Hurt_, Oct 27 2014
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
%H T. D. Noe, <a href="/A002943/b002943.txt">Table of n, a(n) for n = 0..1000</a>
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H Leo Tavares, <a href="/A002943/a002943.jpg">Illustration: Twin Diamond Stars</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KingGraph.html">King Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QueenGraph.html">Queen Graph</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 4*n^2 + 2*n.
%F a(n) = 2*A014105(n). - _Omar E. Pol_, May 21 2008
%F a(n) = floor((2*n + 1/2)^2). - _Reinhard Zumkeller_, Feb 20 2010
%F a(n) = A007494(n) + A173511(n) = A007742(n) + n. - _Reinhard Zumkeller_, Feb 20 2010
%F a(n) = 8*n+a(n-1) - 2 with a(0)=0. - _Vincenzo Librandi_, Jul 20 2010
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Aug 11 2011
%F a(n+1) = A045896(2*n+1). - _Reinhard Zumkeller_, Dec 12 2011
%F G.f.: 2*x*(3+x)/(1-x)^3. - _Colin Barker_, Jan 14 2012
%F 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
%F a(n) = A118729(8*n+5). - _Philippe Deléham_, Mar 26 2013
%F a(n) = 1*A001477(n) + 2*A000217(n) + 3*A000290(n). - _J. M. Bergot_, Apr 23 2014
%F a(n) = 2 * A000217(2*n) = 2 * A014105(n). - _Jon Perry_, Oct 27 2014
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 + log(2)/2 - 1. - _Amiram Eldar_, Feb 22 2022
%F a(n) = A003154(n+1) - A056220(n+1). - _Leo Tavares_, Mar 31 2022
%e 64--65--66--67--68--69--70--71--72
%e |
%e 63 36--37--38--39--40--41--42
%e | | |
%e 62 35 16--17--18--19--20 43
%e | | | | |
%e 61 34 15 4---5---6 21 44
%e | | | | | | |
%e 60 33 14 3 0 7 22 45
%e | | | | | | | |
%e 59 32 13 2---1 8 23 46
%e | | | | | |
%e 58 31 12--11--10---9 24 47
%e | | | |
%e 57 30--29--28--27--26--25 48
%e | |
%e 56--55--54--53--52--51--50--49
%p A002943 := proc(n)
%p 2*n*(2*n+1) ;
%p end proc: # _R. J. Mathar_, Jun 28 2013
%t LinearRecurrence[{3, -3, 1}, {0, 6, 20}, 40] (* _Harvey P. Dale_, Aug 11 2011 *)
%t Table[2 n (2 n + 1), {n, 0, 40}] (* _Harvey P. Dale_, Aug 11 2011 *)
%o (PARI) a(n)=2*n*(2*n+1) \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Magma) [ 4*n^2+2*n: n in [0..50]]; // _Vincenzo Librandi_, Nov 25 2012
%o (Haskell)
%o a002943 n = 2 * n * (2 * n + 1) -- _Reinhard Zumkeller_, Jan 12 2014
%Y Cf. A007742, A033954, A046092, A054000, A014105, A007395, A016813.
%Y Same as A033951 except start at 0.
%Y Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y 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.
%Y 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.
%Y Cf. A003154, A056220.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Formula fixed by _Reinhard Zumkeller_, Apr 09 2010
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