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A230445
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Triangle read by rows: T(n,m) = 3^m*2^(n-m)-1, the number of neighbors in an n-dimensional cubic array.
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1
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0, 1, 2, 3, 5, 8, 7, 11, 17, 26, 15, 23, 35, 53, 80, 31, 47, 71, 107, 161, 242, 63, 95, 143, 215, 323, 485, 728, 127, 191, 287, 431, 647, 971, 1457, 2186, 255, 383, 575, 863, 1295, 1943, 2915, 4373, 6560, 511, 767, 1151, 1727, 2591, 3887, 5831, 8747, 13121
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OFFSET
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0,3
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COMMENTS
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Let n be the dimension of the cubic array.
Let m be the "placement depth" of the cell within the array. m = (number of horizontal or vertical neighbors)-n. 0 <= m <= n.
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.
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.
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
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LINKS
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FORMULA
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T(n,m) = 3^m*2^(n-m)-1, 0 <= m <= n.
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.
T(n,m) = (T(n-1,m) + T(n,m+1))/2, 0 <= m <= n-1.
T(n,m) = 1 + T(n-1,m-1) + T(n,m-1), 1 <= m <= n.
m = T2(n,k)-n, where T2(n,k) is A051162.
G.f. for column m: G(m, x) = x^m*(3^m - 1 - (3^m - 2)*x)/((1 - 2*x)*(1 - x)).
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)).
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EXAMPLE
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Triangle starts:
n \ m 0 1 2 3 4 5 6 7 8 9 10 ...
0: 0
1: 1 2
2: 3 5 8
3: 7 11 17 26
4: 15 23 35 53 80
5: 31 47 71 107 161 242
6: 63 95 143 215 323 485 728
7: 127 191 287 431 647 971 1457 2186
8: 255 383 575 863 1295 1943 2915 4373 6560
9: 511 767 1151 1727 2591 3887 5831 8747 13121 19682
10: 1023 1535 2303 3455 5183 7775 11663 17495 26243 39365 59048
For a 3-d cube, at a corner, the number of horizontal and vertical neighbors is 3, hence m = 3-3 = 0.
Along the edge, the number of horizontal and vertical neighbors is 4, hence m = 4-3 = 1.
In a face, the number of horizontal and vertical neighbors is 5, hence m = 5-3 = 2.
In the interior, the number of horizontal and vertical neighbors is 6, hence m = 6-3 = 3.
T(3,2) = 17 because a cell on the face of a 3-d cube has 17 neighbors.
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MATHEMATICA
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Table[3^m 2^(n - m) - 1, {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Oct 23 2015 *)
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PROG
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(C)
void a10(){int p3[10], p2[10], n, m, a; p3[0]=1; p2[0]=1;
for(n=1; n<10; n++){ p2[n]=p2[n-1]*2; p3[n]=p3[n-1]*3;
for(m=0; m<=d; m++){ a=p3[m]*p2[n-m]-1; printf("%d ", a); }
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CROSSREFS
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Sequence numbers are 1 less than A036561.
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KEYWORD
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AUTHOR
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STATUS
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approved
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