%I #52 Mar 14 2024 18:11:57
%S 1,9,41,109,205,325,473,649,853,1085,1345,1633,1949,2293,2665,3065,
%T 3493,3949,4433,4945,5485,6053,6649,7273,7925,8605,9313,10049,10813,
%U 11605,12425,13273,14149,15053,15985,16945,17933,18949,19993,21065,22165
%N Number of squares on infinite chessboard at <= n knight's moves from a fixed square.
%C Apparently also the number of distinct squares reachable by the (1,3)-leaper in at most n moves. - _R. J. Mathar_, Jan 05 2018
%H Vincenzo Librandi, <a href="/A018836/b018836.txt">Table of n, a(n) for n = 0..1000</a>
%H Erich Friedman, <a href="/A018836/a018836.gif">Illustration of initial terms</a>
%H A. M. Miller and D. L. Farnsworth, <a href="http://dx.doi.org/10.4236/ojdm.2013.33027">Counting the Number of Squares Reachable in k Knight's Moves</a>, Open Journal of Discrete Mathematics, 2013, 3, 151-154
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (1+5*x+12*x^2-8*x^4+4*x^5)*(1+x)/(1-x)^3;
%F a(n) = 1-6*n+14*n^2+4*sign(n*(n-1)*(n-3)). - _Zak Seidov_, Mar 01 2005
%p (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);
%t Table[1-6 n+14 n^2+4 Sign[n(n-1)(n-3)], {n, 0, 50}] (* _Zak Seidov_ *)
%t Join[{1,9,41,109},LinearRecurrence[{3,-3,1},{205,325,473},50]] (* _Harvey P. Dale_, Aug 16 2011 *)
%t 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 *)
%Y Partial sums of A018842. Cf. A098498 (half-infinite board), A001844 (1,1)-leaper, A297740 (2,3)-leaper, A297741 (3,4)-leaper.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_, _Marc LeBrun_