A018842 Number of squares on infinite chessboard at n knight's moves from center.
1, 8, 32, 68, 96, 120, 148, 176, 204, 232, 260, 288, 316, 344, 372, 400, 428, 456, 484, 512, 540, 568, 596, 624, 652, 680, 708, 736, 764, 792, 820, 848, 876, 904, 932, 960, 988, 1016, 1044, 1072, 1100, 1128, 1156
Offset: 0
Links
- Moon Duchin, Counting in Groups: Fine Asymptotic Geometry, Notices of the AMS 63.8 (2016), pp. 871-974. See p. 873.
- Mordechai Katzman, Knight's moves on an infinite board
- M. Katzman, Counting Monomials, J. Alg. Comb. 22 (2005) 331-341.
- A. M. Miller and D. L. Farnsworth, Counting the Number of Squares Reachable in k Knight's Moves, Open Journal of Discrete Mathematics, 2013, 3, 151-154.
- Index entries for linear recurrences with constant coefficients, signature (2,-1).
Programs
-
Maple
(1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1+x)/(1-x)^2; seq(coeff(series(%, x, n+1), x, n), n=0..50);
-
Mathematica
CoefficientList[Series[(1+5x+12x^2-8x^4+4x^5)(1+x)/(1-x)^2, {x,0,50}], x] (* or *) Join[{1,8,32,68,96},LinearRecurrence[{2,-1},{120,148},46]] (* Harvey P. Dale, Jul 05 2011 *)
Formula
a(n) = 28*n-20, n >= 5.
G.f.: (1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1+x)/(1-x)^2.
Extensions
Formula corrected by Jean Drabbe, Mar 11 2013