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.

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

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

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

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

Cf. A209438, A209490, A209491.

Sequence in context: A165899 A316354 A104582 * A133938 A239829 A210034

Adjacent sequences: A209436 A209437 A209438 * A209440 A209441 A209442

Keyword

nonn,tabl

Author

David Nacin, Mar 09 2012

Status

approved