A018840 Number of steps for {2,3} fairy knight to reach (n,0) on infinite chessboard.

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

Offset: 0

Author

Marc LeBrun

Keywords

• nonn
• easy

Comments

This piece is also known as a (2,3)-leaper or a zebra. - Franklin T. Adams-Watters, Dec 27 2017

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1)

Programs

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

Formula

For n >= 18, a(n) = a(n-6) + 2. - David W. Wilson

From Colin Barker, Dec 28 2017: (Start)

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)).

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

(End)

3*a(n) = A004442(n+3)-A084100(n), n>11. - R. J. Mathar, Jan 02 2018