A209437 Table of T(m,n), read by antidiagonals, is the number of subsets of {1,...,n} which contain two elements whose difference is m.

1, 0, 3, 0, 2, 8, 0, 0, 7, 19, 0, 0, 4, 17, 43, 0, 0, 0, 14, 39, 94, 0, 0, 0, 8, 37, 88, 201, 0, 0, 0, 0, 28, 83, 192, 423, 0, 0, 0, 0, 16, 74, 181, 408, 880, 0, 0, 0, 0, 0, 56, 175, 387, 855, 1815, 0, 0, 0, 0, 0, 32, 148, 377, 824, 1775, 3719, 0, 0, 0, 0, 0

Offset

1,3

Comments

m offset is 1, n offset is 2 so 1st entry is T(1,2).

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(m,n) = 2^n - Product_{i=0,...,m-1} F(floor((n + i)/m + 2)) where F(n) is the n-th Fibonacci number.

Example

Table begins:
1, 3, 8, 19, 43, 94, 201, 423, 880, ...
0, 2, 7, 17, 39, 88, 192, 408, 855, ...
0, 0, 4, 14, 37, 83, 181, 387, 824, ...
0, 0, 0,  8, 28, 74, 175, 377, 799, ...
0, 0, 0,  0, 16, 56, 148, 350, 781, ...
0, 0, 0,  0,  0, 32, 112, 296, 700, ...
0, 0, 0,  0,  0,  0,  64, 224, 592, ...
0, 0, 0,  0,  0,  0,   0, 128, 448, ...
0, 0, 0,  0,  0,  0,   0,   0, 256, ...
0, 0, 0,  0,  0,  0,   0,   0,   0, ...
0, 0, 0,  0,  0,  0,   0,   0,   0, ...
.......................................
T(2,3) 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 T(2,3) = 2.

Mathematica

T[m_, n_] := 2^n - Product[Fibonacci[Floor[(n + i)/m + 2]], {i, 0, m - 1}]; Table[T[i, j + 2], {i, 1, 10}, {j, 0, 10}]; Flatten[Table[T[i - j + 1, j + 2], {i, 0, 20}, {j, 0, i}]]

Crossrefs

Cf. A209434, A209435, A209436.

Sequence in context: A302528 A303410 A126671 * A325447 A240660 A347418

Adjacent sequences: A209434 A209435 A209436 * A209438 A209439 A209440

Keyword

nonn,tabl

Author

David Nacin, Mar 09 2012

Status

approved