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A367467
Lexicographically earliest infinite sequence of positive integers such that a(n + a(n)) is distinct for all n.
6
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
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
KEYWORD
nonn
AUTHOR
Neal Gersh Tolunsky, Nov 18 2023
STATUS
approved