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

0,2

Links

Table of n, a(n) for n=0..42.

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).

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.

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 *)

Crossrefs

Cf. A018836 (partial sums), A038522.

Sequence in context: A253295 A290960 A009245 * A139098 A224543 A211633

Adjacent sequences: A018839 A018840 A018841 * A018843 A018844 A018845

Keyword

nonn,nice,walk,easy

Author

N. J. A. Sloane, Marc LeBrun

Extensions

Formula corrected by Jean Drabbe, Mar 11 2013

Status

approved