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Search: id:A126114
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| A126114 |
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Ultimate fixed-point under the mapping n->f(n), where f(n)=n if n is square else f(n)=n-Floor(Sqrt(n)). |
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1
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| 1, 1, 1, 4, 1, 4, 1, 4, 9, 1, 4, 9, 1, 4, 9, 16, 1, 4, 9, 16, 1, 4, 9, 16, 25, 1, 4, 9, 16, 25, 1, 4, 9, 16, 25, 36, 1, 4, 9, 16, 25, 36, 1, 4, 9, 16, 25, 36, 49, 1, 4, 9, 16, 25, 36, 49, 1, 4, 9, 16, 25, 36, 49, 64, 1, 4, 9, 16, 25, 36, 49, 64, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 4, 9, 16, 25
(list; graph; listen)
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OFFSET
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1,4
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FORMULA
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Conjecture. Let t(k) be the largest triangular number t(k)=k(k+1)/2 such that 2t(k) is smaller than n, and denote n-2t(k) by X. Then a(n)=X^2 if X<=k+1, else a(n)=(X-k-1)^2. (This has been verified for n=1,2,3,...,1000.) Illustration. For n=11, we find that 2t(2)=6<11 and 2t(3)=12>11, so that X=11-6=5 and k=2. X>k+1, so we get a(11)=(5-3)^2=4.
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EXAMPLE
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The orbit of 11 under the stated mapping is {11,8,6,4,4,4,4,...} so a(11)=4.
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MATHEMATICA
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f[n_] := FixedPoint[ If[ IntegerQ@ Sqrt@#, #, # - Floor@ Sqrt@# ] &, n]; Array[f, 80] (* Robert G. Wilson v (rgwv@rgwv.com), Mar 08 2007 *)
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CROSSREFS
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Sequence in context: A010685 A099301 A050347 this_sequence A074393 A095666 A089655
Adjacent sequences: A126111 A126112 A126113 this_sequence A126115 A126116 A126117
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KEYWORD
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nonn
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AUTHOR
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John W. Layman (layman(AT)math.vt.edu), Mar 05 2007
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