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A209438 Table of a(d,n) where a(d,n) gives the number of subsets of length n that do not contain an arithmetic progression of length 3 with distance d. 4
1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 7, 4, 2, 1, 1, 13, 8, 4, 2, 1, 1, 24, 16, 8, 4, 2, 1, 1, 44, 28, 16, 8, 4, 2, 1, 1, 81, 49, 32, 16, 8, 4, 2, 1, 1, 149, 91, 64, 32, 16, 8, 4, 2, 1, 1, 274, 169, 112, 64, 32, 16, 8, 4, 2, 1, 1, 504, 312, 196, 128, 64, 32, 16 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
First row just gives the tribonacci numbers.
n offset is 0 but d offset is 1 so 1st entry is a(1,0).
LINKS
FORMULA
a(d,n) = Product_{i=0 to d-1} T(floor(n + i)/d) + 2) where T is the n-th tribonacci number.
EXAMPLE
1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, ...
1, 2, 4, 8, 16, 28, 49, 91, 169, 312, 576, ...
1, 2, 4, 8, 16, 32, 64, 112, 196, 343, 637, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 448, 784, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...
................................................
For a(2,5) 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(2,5) = 2^5-4 = 28.
MATHEMATICA
T[0] = 0; T[1] = 1; T[2] = 1; T[n_] := T[n - 1] + T[n - 2] + T[n - 3]; a[d_, n_] := Product[T[Floor[(n + i)/d] + 2], {i, 0, d - 1}]; Flatten[Table[a[j, i - j], {i, 0, 10}, {j, 0, i}]]
CROSSREFS
Sequence in context: A092479 A124022 A098063 * A106396 A282869 A140998
KEYWORD
nonn,tabl
AUTHOR
David Nacin, Mar 09 2012
STATUS
approved

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Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)