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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

G. C. Greubel, Table of n, a(n) for the first 25 rows, flattened

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

Cf. A209439, A209490, A209491.

Sequence in context: A092479 A124022 A098063 * A106396 A282869 A140998

Adjacent sequences:  A209435 A209436 A209437 * A209439 A209440 A209441

KEYWORD

nonn,tabl

AUTHOR

David Nacin, Mar 09 2012

STATUS

approved

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Last modified November 19 02:32 EST 2018. Contains 317332 sequences. (Running on oeis4.)