A018836
Number of squares on infinite chessboard at <= n knight's moves from a fixed square.
Original entry on oeis.org
1, 9, 41, 109, 205, 325, 473, 649, 853, 1085, 1345, 1633, 1949, 2293, 2665, 3065, 3493, 3949, 4433, 4945, 5485, 6053, 6649, 7273, 7925, 8605, 9313, 10049, 10813, 11605, 12425, 13273, 14149, 15053, 15985, 16945, 17933, 18949, 19993, 21065, 22165
Offset: 0
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Erich Friedman, Illustration of initial terms
- A. M. Miller and D. L. Farnsworth, Counting the Number of Squares Reachable in k Knight's Moves, Open Journal of Discrete Mathematics, 2013, 3, 151-154
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
-
(1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1+x)/(1-x)^3; seq(coeff(series(%, x, n+1), x, n), n=0..50);
-
Table[1-6 n+14 n^2+4 Sign[n(n-1)(n-3)], {n, 0, 50}] (* Zak Seidov *)
Join[{1,9,41,109},LinearRecurrence[{3,-3,1},{205,325,473},50]] (* Harvey P. Dale, Aug 16 2011 *)
CoefficientList[Series[(1 + 5*x + 12*x^2 - 8*x^4 + 4*x^5)*(1 + x)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 26 2012 *)
A038522
On a (2n+1) X (2n+1) board, let m(i) be the number of squares that are i knight's moves from center; sequence gives max m(i) for i >= 0.
Original entry on oeis.org
1, 1, 8, 20, 32, 52, 68, 76, 96, 96, 120, 120, 148, 148, 176, 176, 204, 204, 232, 232, 260, 260, 288, 288, 316, 316, 344, 344, 372, 372, 400, 400, 428, 428, 456, 456, 484, 484, 512, 512, 540, 540, 568, 568, 596, 596, 624, 624, 652, 652, 680, 680, 708, 708
Offset: 0
Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)
On a 5 X 5 board, [ m(0),...,m(4) ]=[ 1,8,8,4,4 ], max=8, so a(2)=8.
- Colin Barker, Table of n, a(n) for n = 0..1000
- Andreas P. Hadjipolakis, Problem E2605, Am. Math. Monthly Vol. 83 (1976), no. 7 (Aug-Sept.), p. 566.
- Roger Weitzenkamp, Solution to Problem E2605: Labels on a Chessboard, Am. Math. Monthly Vol. 84 (1977), p. 822.
- Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
-
LinearRecurrence[{1,1,-1},{1,1,8,20,32,52,68,76,96,96,120,120,148},60] (* Harvey P. Dale, Apr 15 2020 *)
-
Vec((1 + x^2)*(1 + 5*x^2 + 12*x^3 - 4*x^5 + 4*x^6 - 8*x^7 + 4*x^10) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ Colin Barker, Mar 16 2020
A118312
Number of squares on infinite chessboard that a knight can reach in n moves from a fixed square.
Original entry on oeis.org
1, 8, 33, 76, 129, 196, 277, 372, 481, 604, 741, 892, 1057, 1236, 1429, 1636, 1857, 2092, 2341, 2604, 2881, 3172, 3477, 3796, 4129, 4476, 4837, 5212, 5601, 6004, 6421, 6852, 7297, 7756, 8229, 8716, 9217, 9732, 10261, 10804, 11361, 11932, 12517, 13116, 13729, 14356, 14997, 15652
Offset: 0
Anton Chupin (chupin(AT)icmm.ru), May 14 2006
a(2)=33 because knight in 2 moves from square (0,0) can reach one of the following squares: {{0,0}, {-4,-2}, {-4,0}, {-4,2}, {-3,-3}, {-3,-1}, {-3,1}, {-3,3}, {-2,-4}, {-2,0}, {-2,4}, {-1,-3}, {-1,-1}, {-1,1}, {-1,3}, {0,-4}, {0,-2}, {0,2}, {0,4}, {1,-3}, {1,-1}, {1,1}, {1,3}, {2,-4}, {2,0}, {2,4}, {3,-3}, {3,-1}, {3,1}, {3,3}, {4,-2}, {4,0}, {4,2}}.
- M. Petkovic, Mathematics and Chess, Dover Publications (2003), Problem 3.11.
-
I:=[1, 8, 33, 76, 129, 196, 277]; [n le 7 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jul 09 2012
-
Table[ -3 + 4*n + 7*n^2 + 4*Sign[(n - 2)(n - 1)], {n, 0, 100}]
CoefficientList[Series[(1+5*x+12*x^2-8*x^4+4*x^5)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 09 2012 *)
Join[{1,8,33},LinearRecurrence[{3,-3,1},{76,129,196},50]] (* Harvey P. Dale, Dec 05 2014 *)
-
a(n)=7*n^2 + 4*n - 3 + 4*sign((n-2)*(n-1)) \\ Charles R Greathouse IV, Feb 09 2017
Showing 1-3 of 3 results.
Comments