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A265014 Triangle read by rows: T(n,k) = number of neighbors in n-dimensional lattice for generalized neighborhood given with parameter k. 0
2, 4, 8, 6, 18, 26, 8, 32, 64, 80, 10, 50, 130, 210, 242, 12, 72, 232, 472, 664, 728, 14, 98, 378, 938, 1610, 2058, 2186, 16, 128, 576, 1696, 3488, 5280, 6304, 6560, 18, 162, 834, 2850, 6882, 12258, 16866, 19170, 19682, 20, 200, 1160, 4520, 12584, 26024, 41384, 52904, 58024, 59048 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

In an n-dimensional hypercube lattice, the sequence gives the number of nodes situated at a Chebyshev distance of 1 combined with Manhattan distance not greater than k, 1<=k<=n. In terms of cellular automata, it gives the number of neighbors in a generalized neighborhood given with parameter k: at k=1, we obtain von Neumann's neighborhood with 2n neighbors (A005843), and at k=n, we obtain Moore's neighborhood with 3^n-1 neighbors (A024023). It represents partial sums of A013609 rows, first element of each row (equal to 1) excluded.

LINKS

Table of n, a(n) for n=1..55.

D. A. Zaitsev, Generator of lattices

Dmitry Zaitsev, k-neighborhood for Cellular Automata, arXiv preprint arXiv:1605.08870 [cs.DM], 2016.

D. A. Zaitsev, A generalized neighborhood for cellular automata, Theoretical Computer Science, 666 (2017), 21-35.

FORMULA

T(n,k) = Sum_{r=1..k} 2^r*binomial(n,r).

Recurrence: T(n,k) = T(n-1,k-1)-2T(n-1,k-2)+T(n-1,k)+T(n,k-1), T(n,1) = 2n, T(n,n) = 3^n-1.

EXAMPLE

Triangle:

n\k   1    2    3    4    5    6    7    8

--------------------------------------------

1     2

2     4    8

3     6   18   26

4     8   32   64   80

5    10   50  130  210  242

6    12   72  232  472  664  728

7    14   98  378  938 1610 2058 2186

8    16  128  576 1696 3488 5280 6304 6560

...

For instance, for n=3, in a cube:

k=1 corresponds to von Neumann's neighborhood with 6 neighbors situated on facets and given with offsets {(-1,0,0),(1,0,0),(0,-1,0),(0,1,0),(0,0,-1),(0,0,1)};

k=2 corresponds to 18 neighbors situated on facets and sides and given with offsets {(-1,0,0),(1,0,0),(0,-1,0),(0,1,0),(0,0,-1),(0,0,1),(-1,-1,0),(-1,0,-1),(0,-1,-1),(-1,0,1),(-1,1,0),(0,-1,1),(0,1,-1),(1,0,-1),(1,-1,0),(1,1,0),(1,0,1),(0,1,1)};

k=3 corresponds to Moore's neighborhood with 26 neighbors situated on facets, sides and corners given with offsets {(-1,0,0),(1,0,0),(0,-1,0),(0,1,0),(0,0,-1),(0,0,1),(-1,-1,0),(-1,0,-1),(0,-1,-1),(-1,0,1),(-1,1,0),(0,-1,1),(0,1,-1),(1,0,-1),(1,-1,0),(1,1,0),(1,0,1),(0,1,1),(-1,-1,-1),(1,-1,-1),(-1,1,-1),(1,1,-1),(-1,-1,1),(1,-1,1),(-1,1,1),(1,1,1)}.

PROG

(PARI) tabl(nn) = {for (n=1, nn, for (k=1, n, print1(sum(r=1, k, 2^r*binomial(n, r)), ", "); ); print(); ); } \\ Michel Marcus, Dec 16 2015

CROSSREFS

First column equals to A005843.

Diagonal equals to A024023.

Partial row sums of A013609, first element of each row excluded.

Sequence in context: A277331 A124510 A131886 * A262243 A061284 A016017

Adjacent sequences:  A265011 A265012 A265013 * A265015 A265016 A265017

KEYWORD

nonn,tabl

AUTHOR

Dmitry Zaitsev, Nov 30 2015

EXTENSIONS

More terms from Michel Marcus, Dec 16 2015

STATUS

approved

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Last modified June 27 02:59 EDT 2017. Contains 288777 sequences.