A367467 Lexicographically earliest infinite sequence of positive integers such that a(n + a(n)) is distinct for all n.

1, 1, 2, 2, 3, 4, 2, 5, 6, 7, 1, 8, 9, 2, 10, 11, 12, 1, 13, 14, 2, 15, 16, 2, 17, 18, 19, 2, 20, 21, 2, 22, 23, 24, 1, 25, 26, 2, 27, 28, 2, 29, 30, 31, 2, 32, 33, 2, 34, 35, 36, 1, 37, 38, 2, 39, 40, 41, 1, 42, 43, 2, 44, 45, 2, 46, 47, 48, 1, 49, 50, 2, 51, 52, 53, 1, 54, 55, 2, 56, 57, 2, 58, 59, 60, 2, 61, 62, 2

Offset

1,3

Comments

Consider each index i as a location from which one can jump a(i) terms forward. To find a(n) we have to check 2 conditions:

1. The value a(n) can be reached in one jump by at most one previous location.

2. Location n reaches a location in one jump that is not reached in one jump from a location before n.

Described in the above way, the sequence seems to be structured as follows:

A083051 appears to give the indices which cannot be reached from any earlier term; the terms at these indices are 1s and 2s.

A087057 appears to give the indices which can be reached from an earlier term; except for a(2), these terms are first occurrences.

From Thomas Scheuerle, Nov 26 2023: (Start)

Empirical observations:

It appears that this sequence consists of the natural numbers in ascending order interspersed by 1 and 2.

If we consider the distance between successive ones, we will observe a nonperiodic pattern: 9,7,17,17,7,10,7,17,7,10,... . It appears that there are only 7, 10 and 17 with the exception of 9 once.

If we consider the distance between successive twos, we will also observe an interesting nonperiodic pattern: 3,7,7,3,4,3,7,3,4,3,7,7,3,... . It appears that this pattern consists only of 3, 4 and 7. (End)

Links

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..1000

Formula

From Thomas Scheuerle, Nov 26 2023: (Start)

Conjectures:

a(n) = A049472(n) = floor(n*(1 + 1/sqrt(2))) - n, if n is not in A083051.

a(A083051(n)) = A184119(n+1) - A083051(n).

a(a(A083051(n)) + A083051(n)) + a(A083051(n)) + A083051(n) = A328987(n) = floor((a(A083051(n)) + A083051(n))*(1 + 1/sqrt(2))) = floor(A184119(n+1)*(1 + 1/sqrt(2))). (End)

Example

Initial locations and the (by definition) distinct terms that they reach:
     n|  1  2  3  4  5  6  7  8  9
  a(n)|  1  1  2  2  3  4  2  5  6
          =>1=>2====>3
                   ====>4
                      =======>5
                            ====>6
When we evaluate a(i+a(i)) with each index i, we get a distinct value. When i=1, for example, a(1+a(1))=a(1+1)=a(2)=1;  no other i gives 1 as the solution to a(i+a(i)). When i=4, a(4+a(4))=a(4+2)=a(6)=4, and 4 is likewise a solution unique to i=4.

Prog

(MATLAB)
function a = A367467( max_n )
    a = [1 1:2*max_n];
    for n = 3:max_n
        a(n) = 1;
        while consistency(a, n) == false
            a(n) = a(n)+1;
        end
    end
    a = a(1:max_n);
end
function ok = consistency(a, n)
    v = a([1:n] + a(1:n));
    ok = (n == length(unique(v)));
end % Thomas Scheuerle, Nov 21 2023

Crossrefs

Cf. A367832, A293078, A323420, A359807, A367039, A337226, A366691.

Cf. A049472, A083051, A087057, A184119, A328987.

Sequence in context: A209570 A205703 A363100 * A333937 A325785 A207632

Adjacent sequences: A367464 A367465 A367466 * A367468 A367469 A367470

Keyword

nonn

Author

Neal Gersh Tolunsky, Nov 18 2023

Status

approved