cp's OEIS Frontend

A265014 Triangle read by rows: T(n,k) = number of neighbors in n-dimensional lattice for generalized neighborhood given with parameter k.

%I A265014 #39 Sep 26 2018 16:26:09
%S A265014 2,4,8,6,18,26,8,32,64,80,10,50,130,210,242,12,72,232,472,664,728,14,
%T A265014 98,378,938,1610,2058,2186,16,128,576,1696,3488,5280,6304,6560,18,162,
%U A265014 834,2850,6882,12258,16866,19170,19682,20,200,1160,4520,12584,26024,41384,52904,58024,59048
%N A265014 Triangle read by rows: T(n,k) = number of neighbors in n-dimensional lattice for generalized neighborhood given with parameter k.
%C A265014 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.
%H A265014 D. A. Zaitsev, <a href="https://github.com/dazeorgacm/hmn/">Generator of lattices</a>
%H A265014 Dmitry Zaitsev, <a href="https://arxiv.org/abs/1605.08870">k-neighborhood for Cellular Automata</a>, arXiv preprint arXiv:1605.08870 [cs.DM], 2016.
%H A265014 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), 21-35.
%F A265014 T(n,k) = Sum_{r=1..k} 2^r*binomial(n,r).
%F A265014 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.
%e A265014 Triangle:
%e A265014 n\k   1    2    3    4    5    6    7    8
%e A265014 --------------------------------------------
%e A265014 1     2
%e A265014 2     4    8
%e A265014 3     6   18   26
%e A265014 4     8   32   64   80
%e A265014 5    10   50  130  210  242
%e A265014 6    12   72  232  472  664  728
%e A265014 7    14   98  378  938 1610 2058 2186
%e A265014 8    16  128  576 1696 3488 5280 6304 6560
%e A265014 ...
%e A265014 For instance, for n=3, in a cube:
%e A265014 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 A265014 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 A265014 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)}.
%t A265014 T[n_, k_] := 3^n - 2^(k+1) Binomial[n, k+1] Hypergeometric2F1[1, k-n+1, k+2, -2] - 1;
%t A265014 Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 26 2018 *)
%o A265014 (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 A265014 First column equals to A005843.
%Y A265014 Diagonal equals to A024023.
%Y A265014 Partial row sums of A013609, first element of each row excluded.
%K A265014 nonn,tabl
%O A265014 1,1
%A A265014 _Dmitry Zaitsev_, Nov 30 2015
%E A265014 More terms from _Michel Marcus_, Dec 16 2015