Revision History for A140770
(Underlined text is an addition;
strikethrough text is a deletion.)
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#5 by Bruno Berselli at Fri Dec 16 04:12:41 EST 2016
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#4 by Jon E. Schoenfield at Fri Dec 16 03:28:07 EST 2016
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#3 by Jon E. Schoenfield at Fri Dec 16 03:27:55 EST 2016
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| NAME
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3D analog of A081113: the number of (n-1)-step paths a 3D chess king can make starting from one face of the n-by- X n-by- X n cube to the opposite one.
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| COMMENTS
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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
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| EXAMPLE
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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.
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| AUTHOR
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_R. J. Mathar, _, Jun 11 2010
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| STATUS
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approved
editing
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#2 by N. J. A. Sloane at Sun Jul 11 03:00:00 EDT 2010
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| NAME
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A091137(n+1)/A091137(n).
3D analog of A081113: the number of (n-1)-step paths a 3D chess king can make starting from one face of the n-by-n-by-n cube to the opposite one.
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| DATA
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2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2, 64722, 2, 30
1, 16, 289, 4624, 67081, 902500, 11471769, 139570596, 1640493009, 18754206916, 209576262025, 2298031637476, 24798178969729, 263962539461776, 2776718023652329, 28909790108979264, 298278580556192769, 3052900712959977636, 31023767417676585561, 313247762072931012804
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| OFFSET
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0,1
1,2
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| COMMENTS
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2The rule is that the king can move in one step to any of the 26 (=3*3-1) adjacent positions; because we allow only alternatedsolutions with n-1 steps, one component of the direction is enforced and only a denominatorschoice of 9 different next steps remains; if the path is close to the cube Bernoullisurface, even numbersfewer.
It appears that this is identical to A027760. [From Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Sep 24 2008]
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
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| FORMULA
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a(2n)=2. a(2n+1)=A002445(n+1) .
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| EXAMPLE
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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.
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| CROSSREFS
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Cf. A081113.
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| KEYWORD
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nonn,new
nonn
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| AUTHOR
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Paul Curtz (bpcrtz(AT)free.fr), Jul 13 2008
R. J. Mathar, Jun 11 2010
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| EXTENSIONS
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Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 29 2008
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#1 by N. J. A. Sloane at Fri Jan 09 03:00:00 EST 2009
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| NAME
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A091137(n+1)/A091137(n).
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| DATA
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2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2, 64722, 2, 30
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| OFFSET
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0,1
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| COMMENTS
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2 alternated with denominators of Bernoulli numbers.
It appears that this is identical to A027760. [From Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Sep 24 2008]
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| FORMULA
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a(2n)=2. a(2n+1)=A002445(n+1) .
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| KEYWORD
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nonn
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| AUTHOR
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Paul Curtz (bpcrtz(AT)free.fr), Jul 13 2008
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| EXTENSIONS
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Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 29 2008
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| STATUS
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approved
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