A265014 - OEIS

A265014

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

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)}.`
	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 A328964 A061284` `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`