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A209434 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.

%I A209434 #34 Jul 23 2020 18:19:08
%S A209434 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 A209434 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 A209434 75,54,48,32,16,8,4,2,1,233,169,125,81,72,64,32
%N A209434 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 A209434 1st column is the Fibonacci sequence.
%D A209434 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 A209434 G. C. Greubel, <a href="/A209434/b209434.txt">Table of n, a(n) for the first 100 antidiagonals, flattened</a>
%H A209434 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 A209434 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 A209434 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 A209434 Table begins:
%e A209434 1,   1,   1,   1,   1,   1,   1,   1,   1,   1,    1,    ...
%e A209434 2,   2,   2,   2,   2,   2,   2,   2,   2,   2,    2,    ...
%e A209434 3,   4,   4,   4,   4,   4,   4,   4,   4,   4,    4,    ...
%e A209434 5,   6,   8,   8,   8,   8,   8,   8,   8,   8,    8,    ...
%e A209434 8,   9,   12,  16,  16,  16,  16,  16,  16,  16,   16,   ...
%e A209434 13,  15,  18,  24,  32,  32,  32,  32,  32,  32,   32,   ...
%e A209434 21,  25,  27,  36,  48,  64,  64,  64,  64,  64,   64,   ...
%e A209434 34,  40,  45,  54,  72,  96,  128, 128, 128, 128,  128,  ...
%e A209434 55,  64,  75,  81,  108, 144, 192, 256, 256, 256,  256,  ...
%e A209434 89,  104, 125, 135, 162, 216, 288, 384, 512, 512,  512,  ...
%e A209434 144, 169, 200, 225, 243, 324, 432, 576, 768, 1024, 1024, ...
%e A209434 ............................................................
%t A209434 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 A209434 Cf. A209435, A209436, A209437. Columns: A006498, A006500, A031923, A208742, A208743, A009641
%K A209434 nonn,tabl
%O A209434 0,2
%A A209434 _David Nacin_, Mar 09 2012