A049864 -id:A049864 - cp's OEIS frontend

A343017 a(1)=1. For n > 1, a(n) is the n-th positive integer after a(n-1) which cannot be written as a sum of distinct preceding terms in the sequence.

1, 3, 7, 14, 26, 39, 67, 122, 180, 347, 524, 884, 1700, 2564, 4893, 8826, 15593, 28348, 50527, 73536, 136858, 251537, 388362, 662078, 1038501, 1952109, 2983020, 5533878, 8515097, 16211471, 29346362, 45472332, 74818528, 134329628, 251629409, 385580882

Offset: 1

Matthew R. Maas, Apr 02 2021

nonn

Theorem: for n >= 6, a(n) < (Sum_{i=1..n-1} a(i)) - a(n-3). Corollary: a(n) = O(x^n) where x is the positive real solution to x^4 - 2x^3 + x - 1 = 0, exact value (1 + sqrt(3 + 2*sqrt(5)))/2 = 1.86676.... I conjecture that this bound can be tightened further. - Matthew R. Maas, Apr 21 2021

			For a(7):
The first six terms of the sequence are 1, 3, 7, 14, 26, and 39, and the set of all distinct sums of subsets of this set of six numbers is {0 (the empty sum), 1, 3, 4, 7, 8, 10, 11, 14, 15, 17, 18, 21, 22, 24, 25, 26, 27, 29, 30, 33, 34, 36, 37, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 53, 54, 56, 57, 60, 61, 63, 64, 65, 66, 68, 69, 72, 73, 75, 76, 79, 80, 82, 83, 86, 87, 89, 90}. The numbers after 39 which are NOT in this set are {45, 52, 55, 58, 59, 62, 67, ...}. The 7th such number is 67, so a(7)=67.

Rémy Sigrist, Table of n, a(n) for n = 1..41
Rémy Sigrist, C++ program for A343017
Matthew R. Maas, Proof of upper bound on growth rate of a

Cf. A049864. Formula for A049864 was used in calculation of growth rate upper bound. - Matthew R. Maas, Apr 21 2021

// Running set of all possible numbers
// that can be written as a sum of distinct
// terms in the sequence so far. The set
// starts with one element, 0, for the empty
// sum.
HashSet set = new HashSet<>();
set.add(0);
int last = 0;
int nextCount = 1;
while (true) {
    int j = nextCount;
    int position = last;
    while(true) {
        ++position;
        if (set.contains(position)) {
            continue;
        }
        --j;
        if (j == 0) {
            // Output the next term of the sequence.
            System.out.println(position);
            last = position;
            // Add to the running set of possible sums
            // any new sums now possible because of the
            // term just added.
            HashSet setCopy = new HashSet<>(set);
            for (Integer val : setCopy) {
                set.add(Math.addExact(position, val));
            }
            break;
        }
    }
    ++nextCount;
}
(C++) See Links section.

b:= proc(n, i) option remember; n=0 or i>1 and
      b(n, i-1) or i>0 and a(i)<=n and b(n-a(i), i-1)
    end:
a:= proc(n) option remember; local i, j, k; if n=1 then k:= 1
      else k:= a(n-1); for i to n do for j from k+1 while
        b(j, n-1) do od; k:= j od fi; k
    end:
seq(a(n), n=1..20);  # Alois P. Heinz, Apr 02 2021

a[1]=1;a[n_]:=a[n]=Select[Complement[Range[n+Max[l=Union[Total/@Subsets[s=Array[a,n-1]]]]],l],#>Last@s&][[n]];Array[a,20] (* Giorgos Kalogeropoulos, Apr 21 2021 *)

a(25)-a(27) from Alois P. Heinz, Apr 02 2021
a(28)-a(30) from Jinyuan Wang, Apr 02 2021
More terms from Rémy Sigrist, Apr 04 2021

cp's OEIS Frontend

A343017 a(1)=1. For n > 1, a(n) is the n-th positive integer after a(n-1) which cannot be written as a sum of distinct preceding terms in the sequence.

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