%I #81 Oct 17 2022 07:07:07

%S 0,0,0,4,8,16,24,36,48,64,80,100,120,144,168,196,224,256,288,324,360,

%T 400,440,484,528,576,624,676,728,784,840,900,960,1024,1088,1156,1224,

%U 1296,1368,1444,1520,1600,1680,1764,1848,1936,2024,2116,2208,2304,2400,2500,2600,2704,2808

%N Terms in an n X n spiral that do not lie on its principal diagonals.

%C The count of terms not on the principal diagonals is always even.

%C The last digit is the repeating pattern 0,0,0,4,8,6,4,6,8,4, which is palindromic if the leading 0's are removed, 4864684.

%C The sum of the last digits is 40, which is the count of the pattern times 4.

%C A 4 X 4 spiral is the only spiral, aside from a 0 X 0, whose count of terms that do not lie on its principal diagonals equal the count of terms that do [A137932(4) = A042948(4)] making the 4 X 4 the "perfect spiral".

%C Yet another property is mod(a(n), A042948(n)) = 0 iff n is even. This is a large family that includes the 4 X 4 spiral.

%C a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an [n+1] X [n+1] chessboard, when the lone queen is in the most vulnerable position on the board, i.e., on a center square. - _Bob Selcoe_, Feb 12 2015

%C Also the circumference of the (n-1) X (n-1) grid graph for n > 2. - _Eric W. Weisstein_, Mar 25 2018

%C Also the crossing number of the complete bipartite graph K_{5,n}. - _Eric W. Weisstein_, Sep 11 2018

%H Enrique Pérez Herrero, <a href="/A137932/b137932.txt">Table of n, a(n) for n = 0..5000</a>

%H Kival Ngaokrajang, <a href="/A137932/a137932.pdf">Illustration of initial terms</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCircumference.html">Graph Circumference</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCrossingNumber.html">Graph Crossing Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F a(n) = n^2 - (2*n - mod(n,2)) = n^2 - A042948(n).

%F a(n) = 2*A007590(n-1). - _Enrique Pérez Herrero_, Jul 04 2012

%F G.f.: -4*x^3 / ( (1+x)*(x-1)^3 ). a(n) = 4*A002620(n-1). - _R. J. Mathar_, Jul 06 2012

%F From _Bob Selcoe_, Feb 12 2015: (Start)

%F a(n) = (n-1)^2 when n is odd; a(n) = (n-1)^2 - 1 when n is even.

%F a(n) = A002378(n) - A047238(n+1). (End)

%F From _Amiram Eldar_, Mar 20 2022: (Start)

%F Sum_{n>=3} 1/a(n) = Pi^2/24 + 1/4.

%F Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/24 - 1/4. (End)

%F E.g.f.: x*(x - 1)*cosh(x) + (x^2 - x + 1)*sinh(x). - _Stefano Spezia_, Oct 17 2022

%e a(0) = 0^2 - (2(0) - mod(0,2)) = 0.

%e a(3) = 3^2 - (2(3) - mod(3,2)) = 4.

%p A137932:=n->2*floor((n-1)^2/2); seq(A137932(n), n=0..50); # _Wesley Ivan Hurt_, Apr 19 2014

%t Table[2 Floor[(n - 1)^2/2], {n, 0, 20}] (* _Enrique Pérez Herrero_, Jul 04 2012 *)

%t 2 Floor[(Range[20] - 1)^2/2] (* _Eric W. Weisstein_, Sep 11 2018 *)

%t Table[n^2 - 2 n + (1 - (-1)^n)/2, {n, 20}] (* _Eric W. Weisstein_, Sep 11 2018 *)

%t LinearRecurrence[{2, 0, -2, 1}, {0, 0, 4, 8}, 20] (* _Eric W. Weisstein_, Sep 11 2018 *)

%t CoefficientList[Series[-((4 x^2)/((-1 + x)^3 (1 + x))), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 11 2018 *)

%o (Python) a = lambda n: n**2 - (2*n - (n%2))

%o (PARI) A137932(n)={ return(n^2 - (2*n-n%2))} ;

%o print(vector(30,n,A137932(n-1))); /* _R. J. Mathar_, May 12 2014 */

%Y Cf. A042948.

%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.

%K nonn,easy

%O 0,4

%A _William A. Tedeschi_, Feb 29 2008