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A317208
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The Wythoff representation of n: an alternative way of presenting A189921.
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11
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0, 1, 2, 12, 112, 22, 1112, 212, 122, 11112, 2112, 1212, 1122, 222, 111112, 21112, 12112, 11212, 2212, 11122, 2122, 1222, 1111112, 211112, 121112, 112112, 22112, 111212, 21212, 12212, 111122, 21122, 12122, 11222, 2222, 11111112, 2111112, 1211112, 1121112
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OFFSET
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0,3
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COMMENTS
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This is an encoding of the position of n in the A000201, A001950 "Wythoff" table T.
Let T denote the following 3-rowed table, whose rows are n, A = A000201(n), B = A001950(n):
n: 1 2 3 .4 .5 .6 .7 .8 .9 ...
A: 1 3 4 .6 .8 .9 11 12 14 ...
B: 2 5 7 10 13 15 18 20 23 ...
Set a(0)=0. For n>0, locate n in rows A and B of the table, and indicate how to reach that entry starting from column 1. For example, 18 = B(7) = B(B(3)) = B(B(A(2))) = B(B(A(B(1)))), so the path to reach 18 is BBAB, which we write (encoding A as 1, B as 2) as a(18) = 2212.
This is another way of writing the Wythoff representation of n described in Lang (1996) and A189921.
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REFERENCES
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Wolfdieter Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (edts.) Application of Fibonacci numbers vol. 6, Kluwer, Dordrecht, 1996, pp. 319-337.
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LINKS
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MATHEMATICA
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z[n_] := Floor[(n + 1)*GoldenRatio] - n - 1; h[n_] := z[n] - z[n - 1]; w[n_] := Module[{m = n, zm = 0, hm, s = {}}, While[zm != 1, hm = h[m]; AppendTo[s, hm]; If[hm == 1, zm = z[m], zm = z[z[m]]]; m = zm]; s]; a[n_] := FromDigits[ReplaceAll[w[n], {0 :> 2}]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 01 2023 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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