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A171884 Lexicographically earliest injective nonnegative sequence a(n) satisfying |a(n+1) - a(n)| = n for all n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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 [This is correct: by the formula below, a(2*3^k+1...2*3^(k+1)) take all the values in the range [3^(k+1)-1, 5*3^k-2] U [7*3^k-1, 3^(k+2)-2], so the numbers not appearing are those in the range [5*3^k-1, 7*3^k-2] for some k. - Jianing Song, Oct 07 2022]
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 [(i) is true: the indices of records for a(n)/n are n = 1, 2, 3, 4, 6, 7, and 2*3^k+2 for k >= 1, with record values 0, 1/2, 1, 1, 3/2, 7/6, 13/7, and (7*3^k-1)/(2*3^k+2) for k >= 1, so a(n) <= 3.5*n. (ii) needs further justification: the lower natural density is lim_{k->+oo} #{terms <= 7*3^k-2}/(7*3^k-2) = lim_{k->+oo} (4*3^k-1)/(7*3^k-2) = 4/7, and the upper natural density is lim_{k->+oo} #{terms <= 5*3^k-2}/(5*3^k-2) = lim_{k->+oo} (4*3^k-1)/(5*3^k-2) = 4/5. - Jianing Song, Oct 07 2022]
LINKS
R. Munafo, main-A171884.c(C source code to generate the sequence)
FORMULA
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
a(2*3^k + 2*r - 1) = 5*3^k - 1 - r, a(2*3^k + 2*r) = 7*3^k - 2 + r, for k >= 0 and 1 <= r <= 2*3^k. - Jianing Song, Oct 07 2022
EXAMPLE
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.
MATHEMATICA
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)
PROG
(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
a(n) = if(n<=2, n-1, my(k=logint((n-1)\2, 3), r=n-2*3^k); if(r%2, 5*3^k-1-(r+1)/2, 7*3^k-2+r/2)) \\ Jianing Song, Oct 07 2022
CROSSREFS
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
KEYWORD
nonn
AUTHOR
Robert Munafo, Mar 11 2010
EXTENSIONS
Definition edited by M. F. Hasler, Apr 01 2019
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)