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A230445 Triangle read by rows: T(n,m) = 3^m*2^(n-m)-1, the number of neighbors in an n-dimensional cubic array.

%I A230445 #51 Jun 22 2022 23:18:48
%S A230445 0,1,2,3,5,8,7,11,17,26,15,23,35,53,80,31,47,71,107,161,242,63,95,143,
%T A230445 215,323,485,728,127,191,287,431,647,971,1457,2186,255,383,575,863,
%U A230445 1295,1943,2915,4373,6560,511,767,1151,1727,2591,3887,5831,8747,13121
%N A230445 Triangle read by rows: T(n,m) = 3^m*2^(n-m)-1, the number of neighbors in an n-dimensional cubic array.
%C A230445 Let n be the dimension of the cubic array.
%C A230445 Let m be the "placement depth" of the cell within the array. m = (number of horizontal or vertical neighbors)-n. 0 <= m <= n.
%C A230445 Let T(n,m) represent the number of neighbors (horizontally, vertically, or diagonally) a cell has in an n-dimensional cube that has at least 3^n cells.
%C A230445 The sequence forms a triangle structure similar to Pascal’s triangle: T(0,0) in row one, T(1,0), T(1,1) in row two, etc.
%C A230445 The triangle in A094615 is a subtriangle. - _Philippe Deléham_, Oct 31 2013
%C A230445 In a finite n-dimensional hypercube lattice, the sequence gives the number of nodes situated at a Chebyshev distance of 1 for a node, situated on an m-cube bound, which is not on an (m-1)-cube bound. The number of m-cube bounds for n-cube is given by A013609. In cellular automata theory, the cell surrounding with Chebyshev distance 1 is called Moore's neighborhood. For von Neumann neighborhood (with Manhattan distance 1), an analogous sequence is represented by A051162. - _Dmitry Zaitsev_, Oct 22 2015
%H A230445 Dmitry A. Zaitsev, <a href="https://doi.org/10.1016/j.tcs.2016.11.002">A generalized neighborhood for cellular automata</a>, Theoretical Computer Science, 2016, Volume 666, 1 March 2017, Pages 21-35.
%F A230445 T(n,m) = 3^m*2^(n-m)-1, 0 <= m <= n.
%F A230445 T(n,0) = 2^n-1. (A000225)
%F A230445 T(n,n) = 3^n-1. (A024023)
%F A230445 T(n,m) = (3*T(n,m-1)+1)/2, first part of the Collatz sequence for the number 2^n-1, for n >= 1.
%F A230445 T(n,m) = (T(n-1,m) + T(n,m+1))/2, 0 <= m <= n-1.
%F A230445 T(n,m) = 1 + T(n-1,m-1) + T(n,m-1), 1 <= m <= n.
%F A230445 m = T2(n,k)-n, where T2(n,k) is A051162.
%F A230445 From _Wolfdieter Lang_, May 04 2022: (Start)
%F A230445 G.f. for column m: G(m, x) = x^m*(3^m - 1 - (3^m - 2)*x)/((1 - 2*x)*(1 - x)).
%F A230445 G.f. for row polynomials R(n, x) = Sum_{m=1..n} T(n, m)*x^m, for n >= 0: G(z, x) = z*(1 + (2 - 5*z)*x)/((1 - 2*z)*(1 - z)*(1 - 3*x*z)*(1 - x*z)).
%F A230445 (End)
%e A230445 Triangle starts:
%e A230445 n \ m  0    1    2    3    4    5     6     7     8     9    10 ...
%e A230445 0:     0
%e A230445 1:     1    2
%e A230445 2:     3    5    8
%e A230445 3:     7   11   17   26
%e A230445 4:    15   23   35   53   80
%e A230445 5:    31   47   71  107  161  242
%e A230445 6:    63   95  143  215  323  485   728
%e A230445 7:   127  191  287  431  647  971  1457  2186
%e A230445 8:   255  383  575  863 1295 1943  2915  4373  6560
%e A230445 9:   511  767 1151 1727 2591 3887  5831  8747 13121 19682
%e A230445 10: 1023 1535 2303 3455 5183 7775 11663 17495 26243 39365 59048
%e A230445 ... (reformatted (and extended) by _Wolfdieter Lang_, May 04 2022)
%e A230445 For a 3-d cube, at a corner, the number of horizontal and vertical neighbors is 3, hence m = 3-3 = 0.
%e A230445 Along the edge, the number of horizontal and vertical neighbors is 4, hence m = 4-3 = 1.
%e A230445 In a face, the number of horizontal and vertical neighbors is 5, hence m = 5-3 = 2.
%e A230445 In the interior, the number of horizontal and vertical neighbors is 6, hence m = 6-3 = 3.
%e A230445 T(3,2) = 17 because a cell on the face of a 3-d cube has 17 neighbors.
%t A230445 Table[3^m 2^(n - m) - 1, {n, 0, 9}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Oct 23 2015 *)
%o A230445 (C)
%o A230445 void a10(){int p3[10], p2[10], n, m, a; p3[0]=1; p2[0]=1;
%o A230445 for(n=1;n<10;n++){ p2[n]=p2[n-1]*2; p3[n]=p3[n-1]*3;
%o A230445   for(m=0;m<=d;m++){ a=p3[m]*p2[n-m]-1; printf("%d ",a); }
%o A230445   printf("\n"); } } /* _Dmitry Zaitsev_, Oct 23 2015 */
%Y A230445 Cf. A000225, A024023, A051162, A013609.
%Y A230445 Sequence numbers are 1 less than A036561.
%Y A230445 Cf. A048473, A094615.
%K A230445 nonn,easy,tabl
%O A230445 0,3
%A A230445 _Ron R. King_, Oct 18 2013