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%I #28 Mar 06 2018 03:05:18
%S 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,
%T 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,
%U 504,312,196,128,64,32,16,8,4,2,1,927,576,343,256,128
%N 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.
%C First column just gives the tribonacci numbers.
%C n offset is zero, but d offset is one so 1st entry is a(0,1).
%H G. C. Greubel, <a href="/A209439/b209439.txt">Table of n, a(n) for the first 25 rows, flattened</a>
%F T(n,d) = Product_{i=0..d-1} Tr(floor(n + i)/d) + 2) where Tr(n) is the n-th tribonacci number.
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ...
%e 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ...
%e 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 ...
%e 7, 8, 8, 8, 8, 8, 8, 8, 8, 8 ...
%e 13, 16, 16, 16, 16, 16, 16, 16, 16, 16 ...
%e 24, 28, 32, 32, 32, 32, 32, 32, 32, 32 ...
%e 44, 49, 64, 64, 64, 64, 64, 64, 64, 64 ...
%e 81, 91, 112, 128, 128, 128, 128, 128, 128, 128 ...
%e 149, 169, 196, 256, 256, 256, 256, 256, 256, 256 ...
%e 274, 312, 343, 448, 512, 512, 512, 512, 512, 512 ...
%e 504, 576, 637, 784, 1024, 1024, 1024, 1024, 1024, 1024 ...
%e ............................................................
%e 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.
%t 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}]]
%Y Cf. A209438, A209490, A209491.
%K nonn,tabl
%O 0,2
%A _David Nacin_, Mar 09 2012
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