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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.
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%I #35 Apr 09 2023 11:49:58

%S 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,

%T 7,10,13,16,19,18,18,19,19,20,7,11,8,12,14,14,15,9,11,9,12,15,10,14,

%U 16,20,14,17,17,18,22,22,23,22,24,23,23,24,24,25,28,27,22

%N 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.

%C 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

%H Neal Gersh Tolunsky, <a href="/A359034/b359034.txt">Table of n, a(n) for n = 1..10000</a>

%H Thomas Scheuerle, <a href="/A359034/a359034.png">Scatter plot of a(n) per number of groups found. For n = 1...20000</a>. This is the mean number of contiguous terms for each n.

%H Thomas Scheuerle, <a href="/A359034/a359034_1.png">Scatter plot, number of groups of contiguous terms for n = 1...20000</a>

%e 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.

%o (MATLAB)

%o function a = A359034( max_n )

%o a = [1 1];

%o for n = 3:max_n

%o s = 1; e = 1; sm = 1; c = 0;

%o while e < n-1

%o while sm < a(n - 1) && e < (n - 1)

%o e = e + 1; sm = sm + a(e);

%o end

%o if sm == a(n - 1)

%o c = c + (e - s) + 1;

%o end

%o s = s + 1;

%o e = s; sm = a(s);

%o end

%o a(n) = c + 1;

%o end

%o end % _Thomas Scheuerle_, Dec 14 2022

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

%K nonn

%O 1,3

%A _Neal Gersh Tolunsky_, Dec 12 2022