A035008 Total number of possible knight moves on an (n+2) X (n+2) chessboard, if the knight is placed anywhere.

0, 16, 48, 96, 160, 240, 336, 448, 576, 720, 880, 1056, 1248, 1456, 1680, 1920, 2176, 2448, 2736, 3040, 3360, 3696, 4048, 4416, 4800, 5200, 5616, 6048, 6496, 6960, 7440, 7936, 8448, 8976, 9520, 10080, 10656, 11248, 11856, 12480, 13120, 13776

Offset: 0

Author

Ulrich Schimke (ulrschimke(AT)aol.com), Dec 11 1999

Keywords

• easy
• nonn
• nice

Comments

16 times the triangular numbers A000217.

Centered 16-gonal numbers A069129, minus 1. Also, sequence found by reading the segment (0, 16) together with the line from 16, in the direction 16, 48, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Apr 26 2008, Nov 20 2008

For n >= 1, number of permutations of n+1 objects selected from 5 objects v, w, x, y, z with repetition allowed, containing n-1 v's. Examples: at n=1, n-1=0 (i.e., zero v's), and a(1)=16 because we have ww, wx, wy, wz, xw, xx, xy, xz, yw, yx, yy, yz, zw, zx, zy, zz; at n=2, n-1=1 (i.e., one v), and there are 3 permutations corresponding to each one in the n=1 case (e.g., the single v can be inserted in any of three places in the 2-object permutation xy, yielding vxy, xvy, and xyv), so a(2) = 3*a(1) = 3*16 = 48; at n=3, n-1=2 (i.e., two v's), and a(3) = C(4,2)*a(1) = 6*16 = 96; etc. - Zerinvary Lajos, Aug 07 2008 (this needs clarification, Joerg Arndt, Feb 23 2014)

Sequence found by reading the line from 0, in the direction 0, 16, ... and the same line from 0, in the direction 0, 48, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - Omar E. Pol, Oct 03 2011

For n > 0, a(n) is the area of the triangle with vertices at ((n-1)^2, n^2), ((n+1)^2, (n+2)^2), and ((n+3)^2, (n+2)^2). - J. M. Bergot, May 22 2014

For n > 0, a(n) is the number of self-intersecting points in star polygon {4*(n+1)/(2*n+1)}. - Bui Quang Tuan, Mar 28 2015

Equivalently: integers k such that k$ / (k/2)! and k$ / (k/2+1)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information). - Bernard Schott, Dec 02 2021

Examples

			3 X 3-Board: knight can be placed in 8 positions with 2 moves from each, so a(1) = 16.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
• Eric Weisstein's World of Mathematics, Star Polygon.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A002492 (Bishop) and A049450 (Pawn).

Cf. A000217, A069129, A027468, A008586 A038231, A002378, A033996, A046092.

Cf. A348692.

Subsequence of A008586 and of A349081.

Programs

Magma

[8*n*(n+1): n in [0..50]]; // Wesley Ivan Hurt, May 22 2014

Maple

seq(binomial(n+1,2)*4^2, n=0..33); # Zerinvary Lajos, Aug 07 2008

Mathematica

CoefficientList[Series[16 x/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 24 2014 *)
LinearRecurrence[{3,-3,1},{0,16,48},50] (* or *) 16*Accumulate[ Range[ 0,50]] (* Harvey P. Dale, Aug 05 2018 *)

PARI

a(n)=8*n*(n+1) \\ Charles R Greathouse IV, Sep 30 2015

Formula

a(n) = 8*n*(n+1).

G.f.: 16*x/(1-x)^3.

a(n) = A069129(n+1) - 1. - Omar E. Pol, Apr 26 2008

a(n) = binomial(n+1,2)*4^2, n >= 0. - Zerinvary Lajos, Aug 07 2008

a(n) = 8*n^2 + 8*n = 16*A000217(n) = 8*A002378(n) = 4*A046092(n) = 2*A033996(n). - Omar E. Pol, Dec 12 2008

a(n) = a(n-1) + 16*n, with a(0)=0. - Vincenzo Librandi, Nov 17 2010

E.g.f.: 8*exp(x)*x*(2 + x). - Stefano Spezia, May 19 2021

From Amiram Eldar, Feb 22 2023: (Start)

Sum_{n>=1} 1/a(n) = 1/8.

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/8.

Product_{n>=1} (1 - 1/a(n)) = -(8/Pi)*cos(sqrt(3/2)*Pi/2).

Product_{n>=1} (1 + 1/a(n)) = (8/Pi)*cos(Pi/(2*sqrt(2))). (End)

Extensions

More terms from Erich Friedman

Minor errors corrected and edited by Johannes W. Meijer, Feb 04 2010