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%I #24 Oct 23 2022 01:05:09
%S 0,3,2,1,2,3,2,3,4,3,4,5,4,5,6,5,6,7,6,7,8,7,8,9,8,9,10,9,10,11,10,11,
%T 12,11,12,13,12,13,14,13,14,15,14,15,16,15,16,17,16,17,18,17,18,19,18,
%U 19,20,19,20,21,20,21,22,21,22,23,22,23,24,23,24,25
%N a(n) is the least number of knight's moves from corner (0,0) to n-th diagonal of unbounded chessboard.
%C Apart from initial terms, same as A008611. - Anton Chupin, Oct 24 2009
%H G. C. Greubel, <a href="/A047878/b047878.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F a(n) = Min_{i=0..n} A049604(i,n-i).
%F a(3n) = n, a(3n+1) = n+1, a(3n+2) = n+2 for n >= 1.
%F From _Colin Barker_, May 04 2014: (Start)
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.
%F G.f.: x*(3-x-x^2-2*x^3+2*x^4) / ((1-x)^2*(1+x+x^2)). (End)
%F From _Guenther Schrack_, Nov 19 2020: (Start)
%F a(n) = a(n-3) + 1, for n > 4 with a(0) = 0, a(1) = 3, a(2) = 2, a(3) = 1, a(4) = 2;
%F a(n) = (3*n + 6 - 2*(w^(2*n)*(2 + w) + w^n*(1 - w)))/9, for n > 1 with a(0) = 0, a(1) = 3, where w = (-1 + sqrt(-3))/2, a primitive third root of unity;
%F a(n) = (n + 2 - 2*A057078(n))/3 for n > 1;
%F a(n) = A194960(n-2) for n > 2;
%F a(n) = (2*n + 2 - A330396(n))/3 for n > 1. (End)
%t LinearRecurrence[{1,0,1,-1},{0,3,2,1,2,3},80] (* _Harvey P. Dale_, Sep 01 2018 *)
%t Join[{0,3}, Table[(n+2 -2*ChebyshevU[2*n, 1/2])/3, {n,2,75}]] (* _G. C. Greubel_, Oct 22 2022 *)
%o (PARI) concat(0, Vec(x*(2*x^4-2*x^3-x^2-x+3)/((x-1)^2*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, May 04 2014
%o (Magma) I:=[2, 1, 2, 3]; [0,3] cat [n le 4 select I[n] else Self(n-1) +Self(n-3) -Self(n-4): n in [1..81]]; // _G. C. Greubel_, Oct 22 2022
%o (SageMath) (Sage) [0,3]+[(n+2 - 2*chebyshev_U(2*n, 1/2))/3 for n in (2..75)] # _G. C. Greubel_, Oct 22 2022
%Y Cf. A008611, A049604, A057078, A194960, A330396.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_
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