OFFSET
1,3
COMMENTS
Consider each index i as a location from which one can jump a(i) terms forward. No starting index can reach the same value more than once by forward jumps.
The value a(i) at the starting index is not part of the path (and thus allows a(2)=1).
The indices of first occurrences are given by A133263 (essentially triangular numbers + 2).
Changing the definition so that jumps are allowed only from location i to i-a(i) gives A002260.
LINKS
Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, Plot of a(1..100000)^2. It is conjectured that only the topmost straight line in this plot will extend into infinity.
Thomas Scheuerle, Partial view of a(1..80000)^2 equalized to Y axis by shear mapping. This shows the different lengths of the increasing sequences.
Neal Gersh Tolunsky, Ordinal transform of 20000 terms
Neal Gersh Tolunsky, First differences of 20000 terms
FORMULA
a(A133263(n)) = n + 1.
EXAMPLE
We can see, for example, that the terms reachable by jumping forward continuously from i=1 are all distinct (and in this case are just the positive integers):
1, 1, 2, 1, 3, 1, 1, 4, 1, 1, 2, 5
*->1->2---->3------->4---------->5
Beginning at i=9 and jumping forward continuously, we get the sequence 1,2,3,4,5,6,7,9 (in which all terms are likewise distinct).
PROG
(MATLAB)
function a = A367849( max_n )
a = zeros(1, max_n); j = find(a == 0, 1);
while ~isempty(j)
a(j) = 1; k = 1;
if j+k < max_n
while a(j+k) == 0
a(j+k) = k;
if j+2*k-1 < max_n
j = j+(k-1); k = k+1;
else
break;
end
end
end
j = find(a == 0, 1);
end
end % Thomas Scheuerle, Dec 09 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Neal Gersh Tolunsky, Dec 08 2023
STATUS
approved