A018836 Number of squares on infinite chessboard at <= n knight's moves from a fixed square.
1, 9, 41, 109, 205, 325, 473, 649, 853, 1085, 1345, 1633, 1949, 2293, 2665, 3065, 3493, 3949, 4433, 4945, 5485, 6053, 6649, 7273, 7925, 8605, 9313, 10049, 10813, 11605, 12425, 13273, 14149, 15053, 15985, 16945, 17933, 18949, 19993, 21065, 22165
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Erich Friedman, Illustration of initial terms
- 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 (3,-3,1).
Crossrefs
Programs
-
Maple
(1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1+x)/(1-x)^3; seq(coeff(series(%, x, n+1), x, n), n=0..50);
-
Mathematica
Table[1-6 n+14 n^2+4 Sign[n(n-1)(n-3)], {n, 0, 50}] (* Zak Seidov *) Join[{1,9,41,109},LinearRecurrence[{3,-3,1},{205,325,473},50]] (* Harvey P. Dale, Aug 16 2011 *) CoefficientList[Series[(1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1 + x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 26 2012 *)
Formula
G.f.: (1+5*x+12*x^2-8*x^4+4*x^5)*(1+x)/(1-x)^3;
a(n) = 1-6*n+14*n^2+4*sign(n*(n-1)*(n-3)). - Zak Seidov, Mar 01 2005
Comments