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A140770 3D analog of A081113: the number of (n-1)-step paths a 3D chess king can make starting from one face of the n X n X n cube to the opposite one. 0
1, 16, 289, 4624, 67081, 902500, 11471769, 139570596, 1640493009, 18754206916, 209576262025, 2298031637476, 24798178969729, 263962539461776, 2776718023652329, 28909790108979264, 298278580556192769, 3052900712959977636, 31023767417676585561, 313247762072931012804 (list; graph; refs; listen; history; text; internal format)



The rule is that the king can move in one step to any of the 26 (=3*3-1) adjacent positions; because we allow only solutions with n-1 steps, one component of the direction is enforced and only a choice of 9 different next steps remains; if the path is close to the cube surface, even fewer.

This is the square of A081113, because for both x and y coordinates you have A081113(n) possibilities for the path (and you can choose them independently). - Robert Gerbicz, Jun 11 2010


Table of n, a(n) for n=1..20.


Example: for n=2, we can start from any of the 4 places on one face and move from there directly to any of the 4 positions on the opposite side: a(2) = 4*4 = 16.


Cf. A081113.

Sequence in context: A320763 A225194 A027776 * A099279 A202878 A183886

Adjacent sequences:  A140767 A140768 A140769 * A140771 A140772 A140773




R. J. Mathar, Jun 11 2010



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Last modified October 16 05:41 EDT 2021. Contains 348035 sequences. (Running on oeis4.)