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.

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, 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, 75, 54, 48, 32, 16, 8, 4, 2, 1, 233, 169, 125, 81, 72, 64, 32

Offset

0,2

Comments

1st column is the Fibonacci sequence.

References

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.

Links

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened

Katharine A. Ahrens, Combinatorial Applications of the k-Fibonacci Numbers: A Cryptographically Motivated Analysis, Ph. D. thesis, North Carolina State University (2020).

M. Tetiva, Subsets that make no difference d, Mathematics Magazine 84 (2011), no. 4, 300-301.

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.

Example

Table begins:
1,   1,   1,   1,   1,   1,   1,   1,   1,   1,    1,    ...
2,   2,   2,   2,   2,   2,   2,   2,   2,   2,    2,    ...
3,   4,   4,   4,   4,   4,   4,   4,   4,   4,    4,    ...
5,   6,   8,   8,   8,   8,   8,   8,   8,   8,    8,    ...
8,   9,   12,  16,  16,  16,  16,  16,  16,  16,   16,   ...
13,  15,  18,  24,  32,  32,  32,  32,  32,  32,   32,   ...
21,  25,  27,  36,  48,  64,  64,  64,  64,  64,   64,   ...
34,  40,  45,  54,  72,  96,  128, 128, 128, 128,  128,  ...
55,  64,  75,  81,  108, 144, 192, 256, 256, 256,  256,  ...
89,  104, 125, 135, 162, 216, 288, 384, 512, 512,  512,  ...
144, 169, 200, 225, 243, 324, 432, 576, 768, 1024, 1024, ...
............................................................

Mathematica

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}]]

Crossrefs

Cf. A209435, A209436, A209437. Columns: A006498, A006500, A031923, A208742, A208743, A009641

Sequence in context: A347227 A322083 A058399 * A207611 A320973 A058400

Adjacent sequences: A209431 A209432 A209433 * A209435 A209436 A209437

Keyword

nonn,tabl

Author

David Nacin, Mar 09 2012

Status

approved