The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #34 Jul 23 2020 18:19:08
%S 1,2,1,3,2,1,5,4,2,1,8,6,4,2,1,13,9,8,4,2,1,21,15,12,8,4,2,1,34,25,18,
%T 16,8,4,2,1,55,40,27,24,16,8,4,2,1,89,64,45,36,32,16,8,4,2,1,144,104,
%U 75,54,48,32,16,8,4,2,1,233,169,125,81,72,64,32
%N Table T(n,m), read by antidiagonals, is the number of subsets of {1,...,n} which do not contain two elements whose difference is m+1.
%C 1st column is the Fibonacci sequence.
%D M. El-Mikkawy, T. Sogabe, A new family of k-Fibonacci numbers, Appl. Math. Comput. 215 (2010) 4456-4461 doi:10.1016/j.amc.2009.12.069, Table 1.
%H G. C. Greubel, <a href="/A209434/b209434.txt">Table of n, a(n) for the first 100 antidiagonals, flattened</a>
%H Katharine A. Ahrens, <a href="https://repository.lib.ncsu.edu/bitstream/handle/1840.20/37364/etd.pdf">Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis</a>, Ph. D. thesis, North Carolina State University (2020).
%H M. Tetiva, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.4.296">Subsets that make no difference d</a>, Mathematics Magazine 84 (2011), no. 4, 300-301.
%F T(n,m) = Product_{i=0 to m} F(floor[(n + i)/(m + 1) + 2]) where F(n) is the n-th Fibonacci number.
%e Table begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
%e 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
%e 5, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, ...
%e 8, 9, 12, 16, 16, 16, 16, 16, 16, 16, 16, ...
%e 13, 15, 18, 24, 32, 32, 32, 32, 32, 32, 32, ...
%e 21, 25, 27, 36, 48, 64, 64, 64, 64, 64, 64, ...
%e 34, 40, 45, 54, 72, 96, 128, 128, 128, 128, 128, ...
%e 55, 64, 75, 81, 108, 144, 192, 256, 256, 256, 256, ...
%e 89, 104, 125, 135, 162, 216, 288, 384, 512, 512, 512, ...
%e 144, 169, 200, 225, 243, 324, 432, 576, 768, 1024, 1024, ...
%e ............................................................
%t a[n_, m_] := Product[Fibonacci[Floor[(n + i)/(m + 1) + 2]], {i, 0, m}]; Flatten[Table[a[j - i, i], {j, 0, 30}, {i, 0, j}]]
%Y Cf. A209435, A209436, A209437. Columns: A006498, A006500, A031923, A208742, A208743, A009641
%K nonn,tabl
%O 0,2
%A _David Nacin_, Mar 09 2012