A362040 a(n) is the number of distinct sums of one or more contiguous terms in the sequence thus far.

0, 1, 2, 4, 7, 10, 15, 21, 26, 34, 42, 52, 63, 75, 86, 96, 109, 125, 142, 160, 179, 197, 216, 238, 259, 281, 306, 332, 359, 387, 416, 442, 473, 505, 536, 567, 600, 636, 669, 707, 746, 784, 823, 865, 906, 948, 992, 1036, 1083, 1129, 1172, 1222, 1269, 1321, 1374, 1428

Offset: 1

Author

Neal Gersh Tolunsky, Apr 15 2023

Keywords

• nonn

Examples

			At n=1, there are no contiguous subsequences, so a(1)=0.
At n=2, there is one contiguous subsequence: [0], so a(2)=1.
At n=3, there are three contiguous subsequences: [0], [1] and [0, 1], but only two distinct sums (0 and 1), so a(3)=2.
		

Links

• Winston de Greef, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A361798 (number of sums).

Cf. A000217, A002048, A002049.

Programs

Python

from itertools import islice
def gen_a():
    seen = set()
    sums = []
    new = 0
    while True:
        for v in sums: seen.add(v + new)
        sums = [v + new for v in sums]
        sums.append(0)
        new = len(seen)
        yield new
print(list(islice(gen_a(), 60))) # Winston de Greef, Apr 15 2023

Formula

a(n) <= A000217(n).

Extensions

a(13)-a(15) corrected and more terms from Winston de Greef, Apr 15 2023