

A035008


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


37



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
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OFFSET

0,2


COMMENTS

16 times the triangular numbers A000217.
Centered 16gonal 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 n1 v's. Examples: at n=1, n1=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, n1=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 2object permutation xy, yielding vxy, xvy, and xyv), so a(2) = 3*a(1) = 3*16 =48; at n=3, n1=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 18gonal numbers.  Omar E. Pol, Oct 03 2011
For n>0, a(n) represents the area of the triangle with vertices at ((n1)^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 selfintersecting points in star polygon {4*(n+1)/(2*n+1)}.  Bui Quang Tuan, Mar 28 2015


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).


FORMULA

a(n) = 8*n*(n+1).
G.f.: 16*x/(1x)^3.
a(n) = A069129(n+1)  1.  Omar E. Pol, Apr 26 2008
a(n) = C(n+1,2)*4^2, n>=0.  Zerinvary Lajos, Aug 07 2008
a(n) = A000217(n)*16.  Omar E. Pol, Dec 12 2008
a(n) = 8n^2 + 8n = A002378(n)*8 = A046092(n)*4 = A033996(n)*2.  Omar E. Pol, Dec 12 2008
a(n) = a(n1)+16*n, (with a(0)=0).  Vincenzo Librandi, Nov 17 2010


EXAMPLE

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


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 *)


PROG

(MAGMA) [8*n*(n+1): n in [0..50]]; // Wesley Ivan Hurt, May 22 2014
(PARI) a(n)=8*n*(n+1) \\ Charles R Greathouse IV, Sep 30 2015


CROSSREFS

Cf. A033586 (King), A035005 (Queen), A035006 (Rook), A002492 (Bishop) and A049450 (Pawn).
Cf. A000217, A069129, A027468, A008586 A038231, A002378, A033996, A046092.
Sequence in context: A084112 A050428 A134605 * A189972 A023648 A098322
Adjacent sequences: A035005 A035006 A035007 * A035009 A035010 A035011


KEYWORD

easy,nonn,nice


AUTHOR

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


EXTENSIONS

More terms from Erich Friedman
Minor errors corrected and edited by Johannes W. Meijer, Feb 04 2010


STATUS

approved



