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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 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)}. MATHEMATICA T[n_, k_] := 3^n - 2^(k+1) Binomial[n, k+1] Hypergeometric2F1[1, k-n+1, k+2, -2] - 1; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2018 *) 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 October 23 07:20 EDT 2018. Contains 316520 sequences. (Running on oeis4.)