

A171884


Lexicographically earliest injective nonnegative sequence a(n) satisfying a(n+1)  a(n) = n for all n.


4



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

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
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(n1), and thus the ending points (5, 19, 61, ...) by A181655(2n) + A048473(n1).
The first differences have signs (grouped intentionally): +++, , +++, ++++ (5 times ""), +++, +...+ (17 times ""), +++, ... where the number of minus signs is again given by A048473 = A0087761. (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


LINKS

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, mainA171884.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


EXAMPLE

We begin with 0, 0+1=1, 1+2=3. 33=0 cannot be the next term because 0 is already in the sequence so we go to 3+3=6. The next could be 64=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), k1, t=1, 1k, t=k1)) \\ or:
A171884_upto(N, a)=vector(N, k, a+=if(bitand(k, 1)&&k\2!=3^valuation(k(k>1), 3), 1k, k1)) \\ M. F. Hasler, Apr 05 2019


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: A118201 A274647 A113880 * A226940 A098141 A175458
Adjacent sequences: A171881 A171882 A171883 * A171885 A171886 A171887


KEYWORD

nonn


AUTHOR

Robert Munafo, Mar 11 2010


EXTENSIONS

Definition edited by M. F. Hasler, Apr 01 2019


STATUS

approved



