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| A076744
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This sequence with the appropriate denominator (product of (2*3^k-3) k=0..n) produces the expected length of shortest nonintersecting path through n points on a Sierpiński Gasket from corner to corner.
(history;
published version)
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#7 by Joerg Arndt at Thu Mar 15 04:39:02 EDT 2018
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#6 by Jon E. Schoenfield at Wed Mar 14 22:40:45 EDT 2018
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#5 by Jon E. Schoenfield at Wed Mar 14 22:40:41 EDT 2018
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| NAME
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This sequence with the appropriate denominator (product of (2*3^k-3) k=0..n) produces the expected length of shortest non intersectingnonintersecting path through n points on a Sierpiński Gasket from corner to corner.
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approved
editing
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#4 by Jon E. Schoenfield at Fri Aug 14 23:02:35 EDT 2015
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#3 by Jon E. Schoenfield at Fri Aug 14 23:02:33 EDT 2015
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| NAME
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This sequence with the appropriate denominator (product of (2*3^k-3) k=0..n) produces the expected length of shortest non intersecting path through n points on a SierpinskiSierpiński Gasket from corner to corner.
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approved
editing
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#2 by N. J. A. Sloane at Sat Jun 12 03:00:00 EDT 2004
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| MAPLE
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with(combinat, numbcomb): ntop:= := 25: for n from 1 to ntop do a[n]:= ] := sum('numbcomb(n, k)*(-1)^k*1/(2*3^k - 3)', 'k'=0..n): b[n]:= ] := product('2*3^k - 3', 'k'=0..n): od: for n from 1 to ntop do c[n]:=] := solve(x/b[n] = a[n]); od;
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| KEYWORD
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frac,nonn,uned,new
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#1 by N. J. A. Sloane at Fri May 16 03:00:00 EDT 2003
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| NAME
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This sequence with the appropriate denominator (product of (2*3^k-3) k=0..n) produces the expected length of shortest non intersecting path through n points on a Sierpinski Gasket from corner to corner.
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| DATA
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4, 72, 4176, 731808, 381879360, 592267282560, 2733202405059840, 37590062966534453760, 1542797317119230338360320, 189160927199005707074274969600
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| OFFSET
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0,1
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| COMMENTS
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I am looking for the asymptotics of this sequence (scaled by the appropriate denominator). I'm convinced that a(n) / n^(1-log(2)/log(3)) -> constant but need to know more about sequence to solve this problem.
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| MAPLE
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with(combinat, numbcomb): ntop:=25: for n from 1 to ntop do a[n]:= sum('numbcomb(n, k)*(-1)^k*1/(2*3^k - 3)', 'k'=0..n): b[n]:= product('2*3^k - 3', 'k'=0..n): od: for n from 1 to ntop do c[n]:=solve(x/b[n] = a[n]); od;
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| KEYWORD
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frac,nonn,uned
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| AUTHOR
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Nathan B. Shank (nas2(AT)lehigh.edu), Nov 11 2002
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| STATUS
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approved
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