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A171884
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Lexicographically earliest injective nonnegative sequence a(n) satisfying |a(n+1) - a(n)| = n for all n.
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5
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0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 64, 40, 65, 39, 66, 38, 67, 37, 68, 36, 69, 35, 70, 34, 71, 33, 72, 32, 73, 31, 74, 30, 75, 29, 76, 28, 77, 27, 78, 26, 79, 133, 188, 132, 189, 131, 190, 130, 191, 129, 192, 128, 193, 127, 194
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OFFSET
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1,3
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COMMENTS
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The map n -> a(n) is an injective map to the nonnegative integers, i.e., no two terms are identical.
Appears not to contain numbers from the following sets (grouped intentionally): {4, 5}, {14, 15, 16, 17, 18, 19}, {44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61}, etc. The numbers of terms in these groups appears to be A008776. - Paul Raff, Mar 15 2010
The first 23 terms are shared with Recamán's sequence A005132, but from then on they are different. - Philippe Deléham, Mar 01 2013, Omar E. Pol, Jul 01 2013
From M. F. Hasler, May 09 2013:
It appears that the starting points of the gaps (4, 14, 44, 134, 404, 1214, ...) are given by A181655(2n) = A198643(n-1), and thus the ending points (5, 19, 61, ...) by A181655(2n) + A048473(n-1).
The first differences have signs (grouped intentionally): +++, -, +++, -+-+-+-+- (5 times "-"), +++, -+...+- (17 times "-"), +++, ... where the number of minus signs is again given by A048473 = A008776-1. (End)
A correspondent, Dennis Reichard, conjectures that (i) a(n) <= 3.5*n for all n and (ii) the sequence covers 2/3 of all natural numbers. - N. J. A. Sloane, Jun 30 2018
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LINKS
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Table of n, a(n) for n=1..68.
R. Munafo, Lexicographically earliest injective and unbounded sequence A(n) satisfying |A(n+1)-A(n)|=n for all n
R. Munafo, main-A171884.c(C source code to generate the sequence)
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FORMULA
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a(n+1) = a(n) +- n with - iff n is even but not n = 2 + 2*3^k. (Cf. comment from May 09 2013.) - M. F. Hasler, Apr 05 2019
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EXAMPLE
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We begin with 0, 0+1=1, 1+2=3. 3-3=0 cannot be the next term because 0 is already in the sequence so we go to 3+3=6. The next could be 6-4=2 or 6+4=10 but we choose 2 because it is smaller.
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MATHEMATICA
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Contribution from Paul Raff, Mar 15 2010: (Start)
A171884[{}, _, _] := {};
A171884[L_List, max_Integer, True] := If[Length[L] == max, L, With[{n = Length[L]},
If[Last[L] - n < 1 || MemberQ[L, Last[L] - n],
If[MemberQ[L, Last[L] + n],
A171884[Drop[L, -1], max, False],
A171884[Append[L, Last[L] + n], max, True]],
A171884[Append[L, Last[L] - n], max, True]]]]
A171884[L_List, max_Integer, False] := With[{n = Length[L]},
If[MemberQ[L, Last[L] + n],
A171884[Drop[L, -1], max, False],
A171884[Append[L, Last[L] + n], max, True]]]
A171884[{0}, 200, True]
(End)
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PROG
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(PARI) A171884_upto(N, a=0, t=2)=vector(N, k, a+=if(!bitand(k, 1), k-1, t-=1, 1-k, t=k-1)) \\ or:
A171884_upto(N, a)=vector(N, k, a+=if(bitand(k, 1)&&k\2!=3^valuation(k-(k>1), 3), 1-k, k-1)) \\ M. F. Hasler, Apr 05 2019
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CROSSREFS
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Cf. A005132, which allows duplicate values.
Cf. also A118201, in which every value of a(n) and of |a(n+1)-a(n)| occurs exactly once, but does not ensure that the latter is strictly increasing.
Sequence in context: A274647 A113880 A339192 * A339557 A226940 A098141
Adjacent sequences: A171881 A171882 A171883 * A171885 A171886 A171887
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KEYWORD
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nonn
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AUTHOR
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Robert Munafo, Mar 11 2010
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EXTENSIONS
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Definition edited by M. F. Hasler, Apr 01 2019
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STATUS
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approved
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