A209435 Table T(m,n), read by antidiagonals, is the number of subsets of {1,...,n} which do not contain two elements whose difference is m+1.
1, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 4, 6, 8, 1, 2, 4, 8, 9, 13, 1, 2, 4, 8, 12, 15, 21, 1, 2, 4, 8, 16, 18, 25, 34, 1, 2, 4, 8, 16, 24, 27, 40, 55, 1, 2, 4, 8, 16, 32, 36, 45, 64, 89, 1, 2, 4, 8, 16, 32, 48, 54, 75, 104, 144, 1, 2, 4, 8, 16, 32, 64, 72, 81
Offset: 0
Examples
Table begins: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... 1, 2, 4, 6, 9, 15, 25, 40, 64, 104, 169, ... 1, 2, 4, 8, 12, 18, 27, 45, 75, 125, 200, ... 1, 2, 4, 8, 16, 24, 36, 54, 81, 135, 225, ... 1, 2, 4, 8, 16, 32, 48, 72, 108, 162, 243, ... 1, 2, 4, 8, 16, 32, 64, 96, 144, 216, 324, ... 1, 2, 4, 8, 16, 32, 64, 128, 192, 288, 432, ... 1, 2, 4, 8, 16, 32, 64, 128, 256, 384, 576, ... 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 768, ... 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... ................................................
Links
- G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, 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_] := Product[Fibonacci[Floor[(n + i)/(m + 1) + 2]], {i, 0, m}]; Flatten[Table[a[i, j - i], {i, 0, 30}, {j, 0, i}]]
Formula
T(n,m) = Product_{i=0 to m} F(floor[(n + i)/(m + 1) + 2]) where F(n) is the n-th Fibonacci number.
Comments