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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.
4
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
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
KEYWORD
nonn,tabl
AUTHOR
David Nacin, Mar 09 2012
STATUS
approved