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A118966
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a(n) = (n+1)/2 if n occurs among the first n-1 terms of the sequence, otherwise a(n) = 2*n - 1.
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2
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1, 3, 2, 7, 9, 11, 4, 15, 5, 19, 6, 23, 25, 27, 8, 31, 33, 35, 10, 39, 41, 43, 12, 47, 13, 51, 14, 55, 57, 59, 16, 63, 17, 67, 18, 71, 73, 75, 20, 79, 21, 83, 22, 87, 89, 91, 24, 95, 97, 99, 26, 103, 105, 107, 28, 111, 29, 115, 30, 119, 121, 123, 32, 127, 129, 131, 34, 135
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OFFSET
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1,2
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COMMENTS
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Sequence is a permutation of the positive integers. It is also its own inverse (i.e., a(a(n)) = n).
The same sequence can be generated by defining a(0)=0 and a(1)=1 and, for each n>1, choosing the smallest unused positive integer such that max(a(n)/n) will increase or min(a(n)/n) will decrease.
Proof: Three conditions are required to guarantee that the definitions are equivalent. The first condition is that this is a permutation; this is satisfied because this is a permutation involution. This is because (n+1)/2 is the inverse function of 2n-1, which is applied only if n is not already used in the sequence. The second condition is that, with each new term, max(a(n)/n) increases or min(a(n)/n) decreases. This is obviously the case because the next term would be either 2n-1, with would increase max(a(n)/n), or (n+1)/2, which would decrease min(a(n)/n). The third and last condition is that each new term is the smallest possible number satisfying the first two conditions. This holds because 2n-1 is the smallest possible number m*n+b where the slope m is > 1 and a(1) = 1. (A slope > 1 is needed for condition 2.)
(End)
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LINKS
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FORMULA
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MATHEMATICA
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f[s_] := Block[{n = Length@s}, Append[s, If[MemberQ[s, n], (n + 1)/2, 2n - 1]]]; Rest@Nest[f, {1}, 70] (* Robert G. Wilson v, May 16 2006 *)
(* Program to test alternative definition : *)
(* "Permutation of natural number such that max(a(n)/n)-min(a(n)/n) increases monotonously by using smallest possible next number, a(0) = 0, a(1) = 1." *)
Block[{a = {0, 1}, b = {1}, c = {0}, k, r, s}, Do[k = 2; While[Nand[Set[s, Max[#] - Min[#]] > c[[-1]], FreeQ[a, k]] &@ Append[b, Set[r, k/i]], k++]; AppendTo[a, k]; AppendTo[b, r]; AppendTo[c, s], {i, 2, 55}]; a] (* Michael De Vlieger, Dec 11 2020 *)
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PROG
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(MATLAB)
% Program to test alternative definition:
%"Permutation of natural number such that max(a(n)/n)-min(a(n)/n) increases monotonously by using smallest possible next number, a(0) = 0, a(1) = 1."
a(1) = 0;
a(2) = 1;
m_max = 1;
m_min = 1;
n = 3;
t = 1;
while n <= max_n
% search next number t not yet used in a
while ~isempty(find(a==t, 1))
t = t+1;
end
m = t/(n-1);
% check slope m
if m < m_min || m > m_max
% we found a candidate
a(n) = t;
n = n+1;
if m > m_max
m_max = m;
end
if m < m_min
m_min = m;
end
t = 1;
else
% number t does not yet fit
t = t+1;
end
end
end
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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