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%I #97 Jul 10 2022 08:29:15
%S 0,12,40,84,144,220,312,420,544,684,840,1012,1200,1404,1624,1860,2112,
%T 2380,2664,2964,3280,3612,3960,4324,4704,5100,5512,5940,6384,6844,
%U 7320,7812,8320,8844,9384,9940,10512,11100,11704,12324,12960,13612,14280
%N a(n) = 4*n*(2*n + 1).
%C Number of possible king moves on an (n+1) X (n+1) chessboard. E.g., for a 3 X 3 board: king has 4*5 moves, 4*3 moves and 1*8 moves, so a(2)=40. - Ulrich Schimke (ulrschimke(AT)aol.com)
%C Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A085250 in the same spiral. - _Omar E. Pol_, Sep 03 2011
%C Sum of the numbers from 3n to 5n. - _Wesley Ivan Hurt_, Dec 22 2015
%C From _Emeric Deutsch_, Nov 09 2016: (Start)
%C a(n) is the second Zagreb index of the friendship graph F[n]. The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph. The friendship graph (or Dutch windmill graph) F[n] can be constructed by joining n copies of the cycle graph C[3] with a common vertex.
%C For instance, a(2)=40. Indeed, the friendship graph F[2] has 2 edges with end-point degrees 2,2 and 4 edges with end-point degrees 2,4. Then the second Zagreb index is 2*4 + 4*8 = 40. (End)
%C a(n) is the number of vertices in conjoined n X n dodecagons which are arranged into a square array, a.k.a. 3-4-3-12 tiling. - _Donghwi Park_, Dec 20 2020
%D E. Bonsdorff, K. Fabel and O. Riihimaa, Schach und Zahl (Chess and numbers), Walter Rau Verlag, Dusseldorf, 1966.
%H G. C. Greubel, <a href="/A033586/b033586.txt">Table of n, a(n) for n = 0..5000</a>
%H Tim Krabbe, <a href="http://www.xs4all.nl/~timkr/chess2/diary_12.htm">Open Chess Diary, see item 221</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Friendship_graph">Friendship graph</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F Binomial transform of [12, 28, 16, 0, 0, 0, ...] = (12, 40, 84, 144, 220, ...). - _Gary W. Adamson_, Oct 24 2007
%F a(n) = 4 * A014105(n). - _Johannes W. Meijer_, Feb 04 2010
%F a(n) = 16*n + a(n-1) - 4 (with a(0)=0). - _Vincenzo Librandi_, Aug 05 2010
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. - _Harvey P. Dale_, May 10 2011
%F G.f.: 4*x*(3+x)/(1-3*x+3*x^2-x^3). - _Colin Barker_, Jan 06 2012
%F From _Wesley Ivan Hurt_, Feb 25 2014, Dec 22 2015: (Start)
%F a(n) = A008586(n) * A005408(n).
%F a(n) = Sum_{i=3n..5n} i.
%F a(-n) = A085250(n). (End)
%F E.g.f.: (8*x^2 + 12*x)*exp(x). - _G. C. Greubel_, Jul 16 2017
%F From _Vaclav Kotesovec_, Dec 21 2020: (Start)
%F Sum_{n>=1} 1/a(n) = (1 - log(2))/2.
%F Sum_{n>=1} (-1)^n/a(n) = 1/2 - Pi/8 - log(2)/4. (End)
%p A033586:=n->4*n*(2*n+1); seq(A033586(n), n=0..60); # _Wesley Ivan Hurt_, Feb 25 2014
%t Table[4n*(2n + 1), {n, 0, 60}] (* _Stefan Steinerberger_, Apr 08 2006 *)
%t LinearRecurrence[{3,-3,1},{0,12,40},60] (* _Harvey P. Dale_, May 19 2011 *)
%o (PARI) a(n)=4*n*(2*n+1) \\ _Charles R Greathouse IV_, Jul 16 2011
%o (Magma) [4*n*(2*n + 1) : n in [0..50]]; // _Wesley Ivan Hurt_, Dec 22 2015
%Y Cf. A035005 (Queen), A035006 (Rook), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).
%Y Cf. A000217, A005408, A008586, A014105, A085250.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Erich Friedman_
%E Crossref added, minor errors corrected and edited by _Johannes W. Meijer_, Feb 04 2010
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