cp's OEIS Frontend

A266213 Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.

%I A266213 #59 Dec 21 2024 11:08:12
%S A266213 1,1,0,1,2,0,1,4,2,0,1,6,8,2,0,1,8,18,12,2,0,1,10,32,38,16,2,0,1,12,
%T A266213 50,88,66,20,2,0,1,14,72,170,192,102,24,2,0,1,16,98,292,450,360,146,
%U A266213 28,2,0,1,18,128,462,912,1002,608,198,32,2,0
%N A266213 Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)* 2^k.
%C A266213 In an n-dimensional hypercube lattice, the array A(n,r) gives the number of nodes situated at a Manhattan distance equal to r, counting the current node. When counting coordinate offsets for neighboring nodes, binomial(n,k) chooses k nonzero coordinates from n coordinates, binomial(r-1,k-1) partitions the number r as the sum of exactly k nonzero numbers, and 2^k counts combinations of signs for coordinate offsets; starting indexing from 0 adds 1, which counts the current node.
%C A266213 In cellular automata theory, the cell surrounding with Manhattan distance less than or equal to r is called the von Neumann neighborhood of radius r or the diamond-shaped neighborhood to distinguish it from other generalizations of the von Neumann neighborhood for radius r>1, for instance, as a neighborhood having a difference in the range from -r to r in exactly one coordinate (the "narrow" von Neumann neighborhood of radius r).
%C A266213 The square array of partial sums of A(n,r) on rows gives us the Delannoy numbers A008288, which correspond to the number of nodes in the diamond-shaped neighborhood of radius r. - _Dmitry Zaitsev_, Dec 24 2015
%C A266213 For n >= 2, the term A(n,r) gives the number of polyominoes of bounding box 2 x (r+n-1) of area (r + 2(n-1)). Let A'(n,k) be the table A(n,k) without the first two rows. The sum of the terms in the i-th anti-diagonal of A'(n,k) gives the i-th term of A034182. - _Louis Marin_, Dec 11 2024
%H A266213 Vincenzo Librandi, <a href="/A266213/b266213.txt">Table of n, a(n) for n = 0..5150</a>
%H A266213 Bela Bajnok, <a href="https://arxiv.org/abs/1705.07444">Additive Combinatorics: A Menu of Research Problems</a>, arXiv:1705.07444 [math.NT], May 2017. See Sect. 2.3.
%H A266213 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 A266213 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 A266213 A(n, 0)=1, n>=0, A(0, r)=0, r>0.
%F A266213 A(n, r) = A(n, r-1) + A(n-1, r-1) + A(n-1, r).
%F A266213 A(n, r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*binomial(n,k)*2^k.
%F A266213 Triangle T(m, r) = A(m-r, r),  n >= 0, 0 <= r <= n, otherwise 0. - _Wolfdieter Lang_, Jan 31 2016
%F A266213 A(n, k) = [x^k] ((1 + x)/(1 - x))^n. - _Ilya Gutkovskiy_, May 23 2017
%e A266213 The array A(n, k) begins:
%e A266213 n \ k  0  1   2   3    4     5     6      7      8      9
%e A266213 ---------------------------------------------------------
%e A266213 0:     1  0   0   0    0     0     0      0      0      0
%e A266213 1:     1  2   2   2    2     2     2      2      2      2
%e A266213 2:     1  4   8  12   16    20    24     28     32     36
%e A266213 3:     1  6  18  38   66   102   146    198    258    326
%e A266213 4:     1  8  32  88  192   360   608    952   1408   1992
%e A266213 5:     1 10  50 170  450  1002  1970   3530   5890   9290
%e A266213 6:     1 12  72 292  912  2364  5336  10836  20256  35436
%e A266213 7:     1 14  98 462 1666  4942 12642  28814  59906 115598
%e A266213 8:     1 16 128 688 2816  9424 27008  68464 157184 332688
%e A266213 9:     1 18 162 978 4482 16722 53154 148626 374274 864146
%e A266213 ...
%e A266213 For instance, in a 5-hypercube lattice there are 170 nodes situated at a Manhattan distance of 3 for a chosen node.
%e A266213 The triangle T(m, r) begins:
%e A266213 m\r 0  1   2   3   4    5   6   7  8 9 10 ...
%e A266213 0:  1
%e A266213 1:  1  0
%e A266213 2:  1  2   0
%e A266213 3:  1  4   2   0
%e A266213 4:  1  6   8   2   0
%e A266213 5:  1  8  18  12   2    0
%e A266213 6:  1 10  32  38  16    2   0
%e A266213 7:  1 12  50  88  66   20   2   0
%e A266213 8:  1 14  72 170 192  102  24   2  0
%e A266213 9:  1 16  98 292 450  360 146  28  2 0
%e A266213 10: 1 18 128 462 912 1002 608 198 32 2  0
%e A266213 ... Formatted by _Wolfdieter Lang_, Jan 31 2016
%p A266213 # Prints the array by rows.
%p A266213 gf := n -> ((1 + x)/(1 - x))^n: ser := n -> series(gf(n), x, 40):
%p A266213 seq(lprint(seq(coeff(ser(n), x, k), k=0..6)), n=0..9); # _Peter Luschny_, Mar 20 2020
%t A266213 Table[Sum[Binomial[r - 1, k - 1] Binomial[n - r, k] 2^k, {k, 0, Min[n - r, r]}], {n, 0, 10}, {r, 0, n}] // Flatten (* _Michael De Vlieger_, Dec 24 2015 *)
%o A266213 (Python)
%o A266213 from sympy import binomial
%o A266213 def T(n, r):
%o A266213     if r==0: return 1
%o A266213     return sum(binomial(r - 1, k - 1) * binomial(n - r, k) * 2**k for k in range(min(n - r, r) + 1))
%o A266213 for n in range(11): print([T(n, r) for r in range(n + 1)]) # _Indranil Ghosh_, May 23 2017
%Y A266213 Other versions: A035607, A113413, A119800, A122542.
%Y A266213 Partial sums on rows of A give A008288.
%Y A266213 Cf. A001333 (row sums of T). A057077 (alternating row sums of T). - _Wolfdieter Lang_, Jan 31 2016
%K A266213 nonn,tabl
%O A266213 0,5
%A A266213 _Dmitry Zaitsev_, Dec 24 2015