A394252 a(1)=1; for n>1, a(n) is the smallest integer greater than a(n-1) that cannot be written as the sum of exactly n-1 elements from {a(1),...,a(n-1)}, repetition allowed.

1, 2, 5, 10, 28, 56, 112, 326, 652, 1304, 2608, 7712, 15424, 30848, 61696, 182480, 364960, 729920, 1459840, 2919680, 8697344, 17394688, 34789376, 69578752, 139157504, 414552832, 829105664, 1658211328, 3316422656, 6632845312, 13265690624, 39657914368

Offset

1,2

Comments

Among the first 32 computed terms several consecutive doubling blocks occur, for example 1,2; 5,10; 28,56,112; 326,652,1304,2608; 7712,15424,30848,61696; 182480,364960,729920,1459840,2919680. The lengths of these doubling blocks appear to increase over time (2,2,3,4,4,5,...), and the number of blocks of a given length also appears to increase, although with irregular fluctuations. No proof of these structural properties is known to the author.

Empirical observation: a(n+1) appears to be either 2*a(n), or of the form a(n+1) = 3*a(n) - k, where k is the last term of the previous doubling block. No proof of this phenomenon is known to the author.

Links

Table of n, a(n) for n=1..32.

Example

a(1)=1.
For n=2, we must use exactly 1 earlier term {1}.
Representable numbers: 1.
The smallest integer >1 not representable is 2, so a(2)=2.
For n=3, we use exactly 2 terms from {1,2}.
Representable numbers: 2,3,4.
The smallest integer >2 not representable is 5, so a(3)=5.
For n=4, we use exactly 3 terms from {1,2,5}.
Representable numbers: 6,7,8,9.
The smallest integer >5 not representable is 10, so a(4)=10.

Prog

(Python)
from itertools import product
def representable(x, t, A):
    for c in product(A, repeat=t):
        if sum(c)==x:
            return True
    return False
def seq(n):
    A=[1]
    while len(A)<n:
        k=A[-1]+1
        while representable(k, len(A), A):
            k+=1
        A.append(k)
    return A

Crossrefs

Sequence in context: A243797 A120896 A074801 * A324838 A370316 A257889

Adjacent sequences: A394249 A394250 A394251 * A394253 A394254 A394255

Keyword

nonn

Author

Haoqian Wen, Mar 13 2026

Status

approved