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A209439 Table T(n,d) where T(n,d) gives the number of subsets of length n that do not contain an arithmetic progression of length 3 with distance d. 4
1, 2, 1, 4, 2, 1, 7, 4, 2, 1, 13, 8, 4, 2, 1, 24, 16, 8, 4, 2, 1, 44, 28, 16, 8, 4, 2, 1, 81, 49, 32, 16, 8, 4, 2, 1, 149, 91, 64, 32, 16, 8, 4, 2, 1, 274, 169, 112, 64, 32, 16, 8, 4, 2, 1, 504, 312, 196, 128, 64, 32, 16, 8, 4, 2, 1, 927, 576, 343, 256, 128 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
First column just gives the tribonacci numbers.
n offset is zero, but d offset is one so 1st entry is a(0,1).
LINKS
FORMULA
T(n,d) = Product_{i=0..d-1} Tr(floor(n + i)/d) + 2) where Tr(n) is the n-th tribonacci number.
EXAMPLE
1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ...
4, 4, 4, 4, 4, 4, 4, 4, 4, 4 ...
7, 8, 8, 8, 8, 8, 8, 8, 8, 8 ...
13, 16, 16, 16, 16, 16, 16, 16, 16, 16 ...
24, 28, 32, 32, 32, 32, 32, 32, 32, 32 ...
44, 49, 64, 64, 64, 64, 64, 64, 64, 64 ...
81, 91, 112, 128, 128, 128, 128, 128, 128, 128 ...
149, 169, 196, 256, 256, 256, 256, 256, 256, 256 ...
274, 312, 343, 448, 512, 512, 512, 512, 512, 512 ...
504, 576, 637, 784, 1024, 1024, 1024, 1024, 1024, 1024 ...
............................................................
For a(5,2) we count subsets of {1,...,5} that do not contain {1,3,5}, the only d=2 AP possible here. There are 4 subsets containing {1,3,5} so a(5,2) = 2^5-4 = 28.
MATHEMATICA
Trib[0] = 0; Trib[1] = 1; Trib[2] = 1; Trib[n_] := Trib[n - 1] + Trib[n - 2] + Trib[n - 3]; T[n_, d_] := Product[Trib[Floor[(n + i)/d] + 2], {i, 0, d - 1}]; Flatten[Table[T[j - i, i + 1], {j, 0, 15}, {i, 0, j}]]
CROSSREFS
Sequence in context: A165899 A316354 A104582 * A133938 A239829 A210034
KEYWORD
nonn,tabl
AUTHOR
David Nacin, Mar 09 2012
STATUS
approved

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Last modified March 29 08:08 EDT 2024. Contains 371265 sequences. (Running on oeis4.)