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

0,3

Comments

1st row is the Fibonacci sequence.

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.

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, 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, ...
................................................

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}]]

Crossrefs

Cf. A209434, A209436, A209437.

Sequence in context: A308500 A210950 A214314 * A263744 A268956 A208515

Adjacent sequences: A209432 A209433 A209434 * A209436 A209437 A209438

Keyword

nonn,tabl

Author

David Nacin, Mar 09 2012

Status

approved