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A008749
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Expansion of (1+x^6)/((1-x)*(1-x^2)*(1-x^3)).
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2
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1, 1, 2, 3, 4, 5, 8, 9, 12, 15, 18, 21, 26, 29, 34, 39, 44, 49, 56, 61, 68, 75, 82, 89, 98, 105, 114, 123, 132, 141, 152, 161, 172, 183, 194, 205, 218, 229, 242, 255, 268, 281, 296, 309, 324, 339, 354, 369
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OFFSET
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0,3
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COMMENTS
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Conjecture: For n >= 1, A067628(a(n+2)) appears for the first time in A067628. Equivalently, A067628(a(n+2)) is the first T such that the minimal perimeter of polyiamonds of T triangles is a(n+2). - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 05 2002
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LINKS
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FORMULA
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Conjecture: Let b(n>=0) = (0, 1, 1, 1, 1, 3, 1, 3, 3, 3, 3, 5, 3, 5, 5, 5, 5, 7, 3, ...). Equivalently, let b(0) = 0, b(n>=1) = 2*floor((n-1)/6) + 1 + (2 if n+1=0 mod 6; 0 else). Then a(0) = 1, a(n>=1) = a(n-1) + b(n-1). - Winston C. Yang (winston(AT)cs.wisc.edu), Feb 05 2002
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EXAMPLE
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Let n = 8. Then a(n+2) = a(10) = 18. Note A067628(18) = 12 and is the first appearance of 12 in A067628. Equivalently, 12 is the first T such that the min perimeter of polyiamonds of T triangles is 18.
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MATHEMATICA
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CoefficientList[Series[(1+x^6)/((1-x)*(1-x^2)*(1-x^3)), {x, 0, 60}], x] (* G. C. Greubel, Aug 03 2019 *)
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PROG
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(PARI) my(x='x+O('x^60)); Vec((1+x^6)/((1-x)*(1-x^2)*(1-x^3))) \\ G. C. Greubel, Aug 03 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^6)/((1-x)*(1-x^2)*(1-x^3)) )); // G. C. Greubel, Aug 03 2019
(Sage) ((1+x^6)/((1-x)*(1-x^2)*(1-x^3))).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019
(GAP) a:=[1, 1, 2, 3, 4, 5];; for n in [7..60] do a[n]:=a[n-1]+a[n-2]-a[n-4] -a[n-5]+a[n-6]; od; a; # G. C. Greubel, Aug 03 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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