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

Author

David Nacin, Mar 09 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			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.
		

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.

Crossrefs

Cf. A209434, A209435, A209436.

Programs

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}]]

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.