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Number of squares on infinite chessboard at n knight's moves from center.
3

%I #45 Apr 07 2021 14:39:51

%S 1,8,32,68,96,120,148,176,204,232,260,288,316,344,372,400,428,456,484,

%T 512,540,568,596,624,652,680,708,736,764,792,820,848,876,904,932,960,

%U 988,1016,1044,1072,1100,1128,1156

%N Number of squares on infinite chessboard at n knight's moves from center.

%H Moon Duchin, <a href="https://www.ams.org/journals/notices/201608/rnoti-p871.pdf">Counting in Groups: Fine Asymptotic Geometry</a>, Notices of the AMS 63.8 (2016), pp. 871-974. See p. 873.

%H Mordechai Katzman, <a href="http://www.katzman.staff.shef.ac.uk/ComputerAlgebra/knight.ps">Knight's moves on an infinite board</a>

%H M. Katzman, <a href="http://dx.doi.org/10.1007/s108-01-005-4531-6">Counting Monomials</a>, J. Alg. Comb. 22 (2005) 331-341.

%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_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(n) = 28*n-20, n >= 5.

%F G.f.: (1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1+x)/(1-x)^2.

%p (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);

%t 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 *)

%Y Cf. A018836 (partial sums), A038522.

%K nonn,nice,walk,easy

%O 0,2

%A _N. J. A. Sloane_, _Marc LeBrun_

%E Formula corrected by _Jean Drabbe_, Mar 11 2013