A358537 -id:A358537 - cp's OEIS frontend

A359634 a(0)=1 and thereafter a(n) is the length of the longest contiguous group of terms in the sequence thus far that add up to n; if no such group exists, set a(n)=0.

1, 1, 2, 2, 3, 3, 4, 3, 4, 5, 4, 5, 6, 4, 5, 6, 7, 6, 7, 8, 5, 7, 8, 9, 7, 6, 8, 9, 10, 6, 9, 10, 11, 9, 8, 10, 11, 12, 9, 10, 9, 11, 12, 13, 7, 12, 13, 14, 12, 11, 13, 14, 15, 11, 13, 11, 14, 15, 16, 13, 6, 14, 13, 15, 16, 17, 13, 15, 12, 16, 17, 18, 15, 8, 16, 14, 17, 18, 19, 15, 16, 12, 17, 14, 18, 19, 20

Offset: 0

Neal Gersh Tolunsky, Jan 08 2023

nonn

If a zero appears, it is not counted as a term in a contiguous grouping. For example, if (10, 30, 0, 60) is our longest group to sum to 100, this counts as 3 terms, not 4. However, in 50 million terms (computed by Kevin Ryde), a zero has not appeared. Why is this?

How does the lower envelope of this sequence behave?

			a(6) is 4 because in the sequence thus far (1,1,2,2,3,3), the longest run of consecutive terms that sums to 6 is (1,1,2,2), which is 4 terms.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, C program

Cf. A331614, A358537. a(1-16) in A138099 are the same.

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A363279 a(0)=1; a(1)=2. For n>1, a(n) is the number of contiguous groups in the sequence thus far whose sum is n.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 4, 1, 3, 5, 4, 3, 5, 5, 2, 4, 6, 4, 4, 5, 2, 8, 5, 4, 7, 6, 6, 3, 8, 7, 5, 7, 5, 6, 11, 5, 6, 9, 11, 2, 6, 10, 8, 6, 6, 11, 7, 7, 10, 6, 10, 7, 6, 11, 11, 4, 9, 13, 6, 10, 11, 9, 8, 7, 9, 9, 10, 10, 6, 14, 10, 9, 8, 11, 7, 11, 12, 9, 11, 11, 10, 7

Offset: 0

Neal Gersh Tolunsky, May 25 2023

nonn

			a(2)=1 because in the sequence thus far (1, 2), there is only one contiguous subsequence that sums to n=2: (2).
a(7)=3 because in the sequence thus far (1, 2, 1, 2, 1, 1, 2), there are three groups of consecutive terms that sum to n=7: (1, 2, 1, 2, 1); (2, 1, 2, 1, 1); (1, 2, 1, 1, 2).

Neal Gersh Tolunsky, Table of n, a(n) for n = 0..10000
Neal Gersh Tolunsky, Graph of first 10000 terms
Neal Gersh Tolunsky, Graph of first 100000 terms

Cf. A331614, A359634, A358537.

from collections import Counter
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 2]
    sumsn, c =  [2, 3], Counter([1, 2, 3])
    for n in count(2):
        an = c[n]
        yield an
        sumsn = [an] + [s + an for s in sumsn]
        c.update(sumsn)
print(list(islice(agen(), 86))) # Michael S. Branicky, May 25 2023

cp's OEIS Frontend

A359634 a(0)=1 and thereafter a(n) is the length of the longest contiguous group of terms in the sequence thus far that add up to n; if no such group exists, set a(n)=0.

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