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A240917
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a(n) = 2*3^(2*n) - 3*3^n + 1.
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4
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0, 10, 136, 1378, 12880, 117370, 1060696, 9559378, 86073760, 774781930, 6973391656, 62761587778, 564857478640, 5083726873690, 45753570561016, 411782221142578, 3706040248563520, 33354363011912650, 300189269431736776, 2701703431859199778
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OFFSET
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0,2
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COMMENTS
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a(n) is the total number of holes of a triflake-like fractal (fan pattern) after n iterations. The scale factor for this case is 1/3, but for the actual triflake case, it is 1/2, i.e., Sierpiński triangle. The total number of sides is 3*(A198643-1). The perimeter seems to converge to 10/6.
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LINKS
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FORMULA
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a(n) = 9*(a(n-1) + 2*A048473(n-1)) + 1.
a(n) = 1-3^(1+n)+2*9^n.
a(n) = 13*a(n-1)-39*a(n-2)+27*a(n-3).
G.f.: -2*x*(3*x+5) / ((x-1)*(3*x-1)*(9*x-1)). (End).
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MAPLE
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MATHEMATICA
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PROG
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(PARI) a(n)= 2*3^(2*n) - 3*3^n + 1
for(n=0, 100, print1(a(n), ", "))
(PARI) concat(0, Vec(-2*x*(3*x+5)/((x-1)*(3*x-1)*(9*x-1)) + O(x^100))) \\ Colin Barker, Apr 15 2014
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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