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

0,2

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).

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

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 *)

Crossrefs

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

Sequence in context: A095809 A273359 A251422 * A373522 A245932 A373517

Adjacent sequences: A018833 A018834 A018835 * A018837 A018838 A018839

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Marc LeBrun

Status

approved