A018839 -id:A018839 - cp's OEIS frontend

A183043 Triangular array, T(i,j)=number of knight's moves to points on vertical segments (n,0), (n,1),...,(n,n) on infinite chessboard.

0, 3, 2, 2, 1, 4, 3, 2, 3, 2, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 4, 3, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 7, 6, 6, 5, 6, 5, 6, 5, 6, 7, 6, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 6, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 9, 8

Offset: 0

Clark Kimberling, Dec 20 2010

Stated another way, T(n,k) = distance from square (0,0) at center of an infinite open chessboard to square (n,k) via shortest knight path, for 0<=k<=n. - Fred Lunnon, May 18 2014

			Triangle starts:
0,
3,2,
2,1,4,
3,2,3,2,
2,3,2,3,4,
3,4,3,4,3,4,
4,3,4,3,4,5,4,
5,4,5,4,5,4,5,6,
4,5,4,5,4,5,6,5,6,
5,6,5,6,5,6,5,6,7,6
...
See examples under A242511.

Fred Lunnon, Knights in Daze, to appear.

Alois P. Heinz, Rows n = 0..140, flattened
Fred Lunnon, Revised tables & functions for knight's path distance and count (MAGMA code)

Cf. A065775, A183044, A183045, A183046, A018837, A018839, A242511, A242512, A242513, A242514, A242591.

// See link for recursive & explicit algorithms. - Fred Lunnon, May 18 2014

See A065775.

Edited by N. J. A. Sloane, May 23 2014

Offset corrected by Alois P. Heinz, Sep 10 2014

A297740 The number of distinct positions on an infinite chessboard reachable by the (2,3)-leaper in <= n moves.

Original entry on oeis.org

1, 9, 41, 129, 321, 625, 997, 1413, 1885, 2425, 3033, 3709, 4453, 5265, 6145, 7093, 8109, 9193, 10345, 11565, 12853, 14209, 15633, 17125, 18685, 20313, 22009, 23773, 25605, 27505, 29473, 31509, 33613, 35785, 38025, 40333, 42709, 45153, 47665, 50245, 52893

Offset: 0

R. J. Mathar, Jan 05 2018

Colin Barker, Table of n, a(n) for n = 0..1000
R. J. Mathar, Fairy chess leaper minimum moves on the infinite chessboard, (2018).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A018836 (1,2)-leaper or (1,3)-leaper, A297741 (3,4)-leaper.
Partial sums of A018839.
Cf. A253974, A254129, A254459.

LinearRecurrence[{3, -3, 1}, {1, 9, 41, 129, 321, 625, 997, 1413, 1885, 2425}, 50] (* Paolo Xausa, Mar 17 2024 *)

Vec((1 + x)*(1 + 5*x + 12*x^2 + 20*x^3 + 28*x^4 - 20*x^5 - 24*x^6 + 12*x^8) / (1 - x)^3 + O(x^40)) \\ Colin Barker, Jan 07 2018

a(n) = 34*n^2 + 30*n + 9 for n >= 6.
From Colin Barker, Jan 05 2018: (Start)
G.f.: (1 + x)*(1 + 5*x + 12*x^2 + 20*x^3 + 28*x^4 - 20*x^5 - 24*x^6 + 12*x^8) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>9. (End)

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A183043 Triangular array, T(i,j)=number of knight's moves to points on vertical segments (n,0), (n,1),...,(n,n) on infinite chessboard.

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