A209436 Table of a(n,m) = number of subsets of {1,...,n} which contain two elements whose difference is m+1.
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, 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, 181, 74, 16, 0, 0, 0, 0, 0, 0, 1815, 855, 387, 175, 56, 0, 0, 0, 0
Offset: 0
Examples
Table begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
8, 7, 4, 0, 0, 0, 0, 0, 0, 0, 0, ...
19, 17, 14, 8, 0, 0, 0, 0, 0, 0, 0, ...
43, 39, 37, 28, 16, 0, 0, 0, 0, 0, 0, ...
94, 88, 83, 74, 56, 32, 0, 0, 0, 0, 0, ...
201, 192, 181, 175, 148, 112, 64, 0, 0, 0, 0, ...
423, 408, 387, 377, 350, 296, 224, 128, 0, 0, 0, ...
880, 855, 824, 799, 781, 700, 592, 448, 256, 0, 0, ...
......................................................
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.
Links
- G. C. Greubel, Table of n, a(n) for the first 100 aintidiagonals, flattened
- M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301.
Programs
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Mathematica
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}]]
Formula
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.