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A061709
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Consider a (hollow) triangle with n cells on each edge, for a total of 3(n-1) cells if n>1, or 1 cell if n=1; a(n) is number of ways of labeling cells with 0's and 1's; triangle may be rotated and turned over.
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2
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1, 4, 20, 104, 752, 5600, 44224, 350592, 2800384, 22377984, 178990080, 1431721984, 11453509632, 91626496000, 733009854464, 5864066220032, 46912512917504, 375300002545664, 3002399885885440, 24019198281252864, 192153585175232512, 1537228674957312000
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = (1/6)*(2^(3*(n-1))+2^n+3*2^(floor((3*n-1)/2))) for n>1.
a(2)=4, a(3)=20, a(4)=104, a(5)=752, a(n)=10*a(n-1)-8*a(n-2)- 80*a(n-3)+ 128*a(n-4). - Harvey P. Dale, Apr 22 2013
G.f.: -x*(64*x^4+16*x^3-12*x^2-6*x+1) / ((2*x-1)*(8*x-1)*(8*x^2-1)). - Colin Barker, Mar 17 2015
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EXAMPLE
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a(2) = 4, the labelings being {000}, {001}, 011}, {111}. Some of the 20 solutions for n=3 are as follows:
..0......1.......0......1.......1.......1.......0
.0.0....0.0.....1.0....1.0.....0.0.....0.0.....1.1
0.0.0..0.0.0...0.0.0..0.0.0...1.0.0...0.1.0...0.0.0
The first solution for n = 4 is
...0
..0.0
.0...0
0.0.0.0
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MATHEMATICA
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Join[{1}, Table[((2^(3(n-1)))+2^n+3*2^Floor[(3n-1)/2])/6, {n, 2, 30}]] (* or *) Join[{1}, LinearRecurrence[{10, -8, -80, 128}, {4, 20, 104, 752}, 30]] (* Harvey P. Dale, Apr 22 2013 *)
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PROG
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(PARI) Vec(-x*(64*x^4+16*x^3-12*x^2-6*x+1)/((2*x-1)*(8*x-1)*(8*x^2-1)) + O(x^100)) \\ Colin Barker, Mar 17 2015
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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