A359034 a(n+1) is the sum of the number of terms in all groups of contiguous terms that add up to a(n); a(1)=1.

1, 1, 2, 3, 3, 4, 4, 5, 3, 5, 4, 6, 6, 7, 7, 8, 10, 11, 4, 7, 9, 9, 10, 12, 13, 14, 13, 15, 8, 11, 7, 10, 13, 16, 19, 18, 18, 19, 19, 20, 7, 11, 8, 12, 14, 14, 15, 9, 11, 9, 12, 15, 10, 14, 16, 20, 14, 17, 17, 18, 22, 22, 23, 22, 24, 23, 23, 24, 24, 25, 28, 27, 22

Offset: 1

Author

Neal Gersh Tolunsky, Dec 12 2022

Keywords

• nonn

Comments

If strongly smoothened, this sequence displays growth. This growth appears to be caused by the number of groups which is increasing by growing length of the sequence roughly proportional to n^(1/2). But the length of the groups appears to be nearly uninfluenced by this. - Thomas Scheuerle, Dec 14 2022

Examples

			a(17) is 10 because in the sequence so far (1, 1, 2, 3, 3, 4, 4, 5, 3, 5, 4, 6, 6, 7, 7, 8), these are the ways of adding contiguous terms to get a(16)=8: (2, 3, 3); (4, 4); (5, 3); (3, 5); (8). This is 10 terms in total, so a(17) is 10. Notice groups (5,3) and (3,5) overlap.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
• Thomas Scheuerle, Scatter plot of a(n) per number of groups found. For n = 1...20000. This is the mean number of contiguous terms for each n.
• Thomas Scheuerle, Scatter plot, number of groups of contiguous terms for n = 1...20000

Crossrefs

Cf. A331614, A358919. Begins the same as A124056 (until a(13)).

Programs

MATLAB

function a = A359034( max_n )
    a = [1 1];
    for n = 3:max_n
        s = 1; e = 1; sm = 1; c = 0;
        while e < n-1
            while sm < a(n - 1) && e < (n - 1)
                e = e + 1; sm = sm + a(e);
            end
            if sm == a(n - 1)
                c = c + (e - s) + 1;
            end
            s = s + 1;
            e = s; sm = a(s);
        end
        a(n) = c + 1;
    end
end % Thomas Scheuerle, Dec 14 2022