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

Author

N. J. A. Sloane, Marc LeBrun

Keywords

• nonn
• nice
• easy

Comments

Apparently also the number of distinct squares reachable by the (1,3)-leaper in at most n moves. - R. J. Mathar, Jan 05 2018

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

Partial sums of A018842. Cf. A098498 (half-infinite board), A001844 (1,1)-leaper, A297740 (2,3)-leaper, A297741 (3,4)-leaper.

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