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Number of steps for {2,3} fairy knight to reach (n,0) on infinite chessboard.
2

%I #29 Jan 05 2018 07:35:31

%S 0,5,4,5,2,5,2,5,4,5,4,7,4,5,6,7,6,7,6,7,8,9,8,9,8,9,10,11,10,11,10,

%T 11,12,13,12,13,12,13,14,15,14,15,14,15,16,17,16,17,16,17,18,19,18,19,

%U 18,19,20,21,20,21,20,21,22,23,22,23,22,23,24,25,24,25,24,25,26,27,26,27,26,27

%N Number of steps for {2,3} fairy knight to reach (n,0) on infinite chessboard.

%C This piece is also known as a (2,3)-leaper or a zebra. - _Franklin T. Adams-Watters_, Dec 27 2017

%C Apparently also the minimum number of moves of the (1,5)-leaper to reach (n,n) starting from (0,0). - _R. J. Mathar_, Jan 05 2018

%H Colin Barker, <a href="/A018840/b018840.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1)

%F For n >= 18, a(n) = a(n-6) + 2. - _David W. Wilson_

%F From _Colin Barker_, Dec 28 2017: (Start)

%F G.f.: x*(5 - x + x^2 - 3*x^3 + 3*x^4 - 3*x^5 - 2*x^6 + 2*x^9 - 2*x^12 + 2*x^13 - 2*x^16 + 2*x^17) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).

%F a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.

%F (End)

%F 3*a(n) = A004442(n+3)-A084100(n), n>11. - _R. J. Mathar_, Jan 02 2018

%o (PARI) concat(0, Vec(x*(5 - x + x^2 - 3*x^3 + 3*x^4 - 3*x^5 - 2*x^6 + 2*x^9 - 2*x^12 + 2*x^13 - 2*x^16 + 2*x^17) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^100))) \\ _Colin Barker_, Dec 28 2017

%K nonn,easy

%O 0,2

%A _Marc LeBrun_