%I
%S 2,4,8,6,18,26,8,32,64,80,10,50,130,210,242,12,72,232,472,664,728,14,
%T 98,378,938,1610,2058,2186,16,128,576,1696,3488,5280,6304,6560,18,162,
%U 834,2850,6882,12258,16866,19170,19682,20,200,1160,4520,12584,26024,41384,52904,58024,59048
%N Triangle read by rows: T(n,k) = number of neighbors in ndimensional lattice for generalized neighborhood given with parameter k.
%C In an ndimensional 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^n1 neighbors (A024023). It represents partial sums of A013609 rows, first element of each row (equal to 1) excluded.
%H D. A. Zaitsev, <a href="https://github.com/dazeorgacm/hmn/">Generator of lattices</a>
%H Dmitry Zaitsev, <a href="https://arxiv.org/abs/1605.08870">kneighborhood for Cellular Automata</a>, arXiv preprint arXiv:1605.08870 [cs.DM], 2016.
%H D. A. Zaitsev, <a href="https://doi.org/10.1016/j.tcs.2016.11.002">A generalized neighborhood for cellular automata</a>, Theoretical Computer Science, 666 (2017), 2135.
%F T(n,k) = Sum_{r=1..k} 2^r*binomial(n,r).
%F Recurrence: T(n,k) = T(n1,k1)2T(n1,k2)+T(n1,k)+T(n,k1), T(n,1) = 2n, T(n,n) = 3^n1.
%e Triangle:
%e n\k 1 2 3 4 5 6 7 8
%e 
%e 1 2
%e 2 4 8
%e 3 6 18 26
%e 4 8 32 64 80
%e 5 10 50 130 210 242
%e 6 12 72 232 472 664 728
%e 7 14 98 378 938 1610 2058 2186
%e 8 16 128 576 1696 3488 5280 6304 6560
%e ...
%e For instance, for n=3, in a cube:
%e 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)};
%e 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)};
%e 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)}.
%o (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
%Y First column equals to A005843.
%Y Diagonal equals to A024023.
%Y Partial row sums of A013609, first element of each row excluded.
%K nonn,tabl
%O 1,1
%A _Dmitry Zaitsev_, Nov 30 2015
%E More terms from _Michel Marcus_, Dec 16 2015
