A137932 Terms in an n X n spiral that do not lie on its principal diagonals.

0, 0, 0, 4, 8, 16, 24, 36, 48, 64, 80, 100, 120, 144, 168, 196, 224, 256, 288, 324, 360, 400, 440, 484, 528, 576, 624, 676, 728, 784, 840, 900, 960, 1024, 1088, 1156, 1224, 1296, 1368, 1444, 1520, 1600, 1680, 1764, 1848, 1936, 2024, 2116, 2208, 2304, 2400, 2500, 2600, 2704, 2808

Offset: 0

Author

William A. Tedeschi, Feb 29 2008

Keywords

• nonn
• easy

Comments

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

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.

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

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".

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.

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

Also the circumference of the (n-1) X (n-1) grid graph for n > 2. - Eric W. Weisstein, Mar 25 2018

Also the crossing number of the complete bipartite graph K_{5,n}. - Eric W. Weisstein, Sep 11 2018

Examples

			a(0) = 0^2 - (2(0) - mod(0,2)) = 0.
a(3) = 3^2 - (2(3) - mod(3,2)) = 4.
		

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 0..5000
• Kival Ngaokrajang, Illustration of initial terms.
• Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
• Eric Weisstein's World of Mathematics, Graph Circumference.
• Eric Weisstein's World of Mathematics, Graph Crossing Number.
• Eric Weisstein's World of Mathematics, Grid Graph.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A042948.

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.

Programs

Maple

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

Mathematica

Table[2 Floor[(n - 1)^2/2], {n, 0, 20}] (* Enrique Pérez Herrero, Jul 04 2012 *)
2 Floor[(Range[20] - 1)^2/2] (* Eric W. Weisstein, Sep 11 2018 *)
Table[n^2 - 2 n + (1 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Sep 11 2018 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 0, 4, 8}, 20] (* Eric W. Weisstein, Sep 11 2018 *)
CoefficientList[Series[-((4 x^2)/((-1 + x)^3 (1 + x))), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 11 2018 *)

PARI

A137932(n)={ return(n^2 - (2*n-n%2))} ;
print(vector(30,n,A137932(n-1))); /* R. J. Mathar, May 12 2014 */

Python

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

Formula

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

a(n) = 2*A007590(n-1). - Enrique Pérez Herrero, Jul 04 2012

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

From Bob Selcoe, Feb 12 2015: (Start)

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

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

From Amiram Eldar, Mar 20 2022: (Start)

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

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

E.g.f.: x*(x - 1)*cosh(x) + (x^2 - x + 1)*sinh(x). - Stefano Spezia, Oct 17 2022