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A209436 Table of a(n,m) = number of subsets of {1,...,n} which contain two elements whose difference is m+1.

%I A209436 #25 Jan 03 2018 15:57:20
%S A209436 0,0,0,1,0,0,3,0,0,0,8,2,0,0,0,19,7,0,0,0,0,43,17,4,0,0,0,0,94,39,14,
%T A209436 0,0,0,0,0,201,88,37,8,0,0,0,0,0,423,192,83,28,0,0,0,0,0,0,880,408,
%U A209436 181,74,16,0,0,0,0,0,0,1815,855,387,175,56,0,0,0,0
%N A209436 Table of a(n,m) = number of subsets of {1,...,n} which contain two elements whose difference is m+1.
%H A209436 G. C. Greubel, <a href="/A209436/b209436.txt">Table of n, a(n) for the first 100 aintidiagonals, flattened</a>
%H A209436 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 A209436 a(n,m) = 2^n - Product_{i=0 to m} F(floor[(n + i)/(m + 1) + 2]) where F(n) is the n-th Fibonacci number.
%e A209436 Table begins:
%e A209436 0,   0,   0,   0,   0,   0,   0,   0,   0,   0, 0, ...
%e A209436 0,   0,   0,   0,   0,   0,   0,   0,   0,   0, 0, ...
%e A209436 1,   0,   0,   0,   0,   0,   0,   0,   0,   0, 0, ...
%e A209436 3,   2,   0,   0,   0,   0,   0,   0,   0,   0, 0, ...
%e A209436 8,   7,   4,   0,   0,   0,   0,   0,   0,   0, 0, ...
%e A209436 19,  17,  14,  8,   0,   0,   0,   0,   0,   0, 0, ...
%e A209436 43,  39,  37,  28,  16,  0,   0,   0,   0,   0, 0, ...
%e A209436 94,  88,  83,  74,  56,  32,  0,   0,   0,   0, 0, ...
%e A209436 201, 192, 181, 175, 148, 112, 64,  0,   0,   0, 0, ...
%e A209436 423, 408, 387, 377, 350, 296, 224, 128, 0,   0, 0, ...
%e A209436 880, 855, 824, 799, 781, 700, 592, 448, 256, 0, 0, ...
%e A209436 ......................................................
%e A209436 a(3,1) is the number of subsets of {1,2,3} containing two elements whose difference is two.  There are 2 of these: {1,3} and {1,2,3} so a(1,3) = 2.
%t A209436 a[n_, m_] := 2^n - Product[Fibonacci[Floor[(n + i)/(m + 1) + 2]], {i, 0, m}]; Flatten[Table[a[j - i, i], {j, 0, 20}, {i, 0, j}]]
%Y A209436 Cf. A209434, A209435, A209437.
%K A209436 nonn,tabl
%O A209436 0,7
%A A209436 _David Nacin_, Mar 09 2012