A358537 For n > 0, a(n) is the total number of terms in all contiguous subsequences of the terms up to a(n-1) that sum to n; a(0) = 1.

1, 1, 2, 2, 5, 4, 4, 2, 2, 5, 7, 8, 6, 11, 10, 16, 5, 22, 6, 19, 15, 22, 20, 9, 18, 5, 14, 16, 23, 9, 8, 11, 16, 12, 19, 21, 0, 21, 8, 20, 11, 17, 25, 28, 4, 18, 4, 30, 23, 40, 7, 20, 18, 18, 14, 9, 40, 9, 29, 32, 23, 6, 17, 23, 16, 8, 26, 32, 35, 27, 64, 10

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Example

To find a(4), we look at the sequence so far (1, 1, 2, 2) to find contiguous subsequences that sum to 4: (1, 1, 2) and (2, 2). This is five terms in total, so a(4) = 5. Notice that the two subsequences overlap.
a(40) is 11 because the following contiguous subsequences sum to 40: (6, 19, 15); (23, 9, 8); (19, 21); (19, 21, 0). This is a total of 11 terms.

Maple

N:= 100: V:= Array(0..N):
V[0]:= 1:
for n from 0 to N-1 do
  s:= 0;
  for j from n to 0 by -1 do
    s:= s + V[j];
    if s > N then break fi;
    if s > n then V[s]:= V[s] + n-j+1 fi;
  od;
od:
convert(V, list); # Robert Israel, Feb 16 2023

Prog

(PARI) { for (n=1, #a=m=vector(72), print1 (a[n] = if (n==1, 1, m[n-1])", "); s = w = 0; forstep (k=n, 1, -1, w++; if ((s += a[k]) > #m, break, s, m[s] += w))) } \\ Rémy Sigrist, Feb 09 2023

Crossrefs

Cf. A359034, A331614, A359634.

Sequence in context: A228526 A209745 A249620 * A025498 A193693 A128971

Adjacent sequences: A358534 A358535 A358536 * A358538 A358539 A358540

Keyword

nonn,look

Author

Neal Gersh Tolunsky, Dec 18 2022

Extensions

Data edited by Yifan Xie, Feb 08 2023

More terms from Rémy Sigrist, Feb 09 2023

Status

approved