A244750 0-additive sequence: a(n) is the smallest number larger than a(n-1) which is not the sum of any subset of earlier terms, with initial values {0, 2, 3, 4}.

0, 2, 3, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184

Offset

1,2

Comments

0, 2, 3 and then terms in A000079 > 3. - Michael S. Branicky, Nov 07 2025

Links

Table of n, a(n) for n=1..36.

Steven R. Finch, Are 0-additive sequences always regular?, Amer. Math. Monthly, 99 (1992), 671-673.

R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)

Formula

a(n) = 2*a(n-1) = 2^(n-2) for n > 4. - Michael S. Branicky, Nov 07 2025

Example

a(5) cannot be 5=2+3. It cannot be 6=2+4. It cannot be 7=3+4, and becomes a(5)=8.
a(6) cannot be 9=2+3+4. It cannot be 10=2+8. It cannot be 11=3+8. It cannot be 12 = 4+8. It cannot be 13=2+3+8. It cannot be 14=2+4+8. It cannot be 15=3+4+8, and becomes a(6)=16.

Maple

A244750:= proc(n)
    option remember;
    if n <= 4 then
        op(n, [0, 2, 3, 4]);
    else
        prev := {seq(procname(k), k=1..n-1)} ;
        for a from procname(n-1)+1 do
            awrks := true ;
            for asub in combinat[choose](prev) do
                if add(p, p=asub) = a then
                    awrks := false;
                    break;
                end if;
            end do:
            if awrks then
                return a;
            end if;
        end do:
    end if;
end proc:
for n from 1 do
    print(A244750(n)) ;
end do: # R. J. Mathar, Jul 12 2014

Mathematica

 f[s_List] := f[n] = Block[{k = s[[-1]] + 1, ss = Union[Plus @@@ Subsets[s]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[ f[#] &, {0, 2, 3, 4}, 16]

Prog

(Python)
from itertools import count, islice
def agen(): # generator of terms
    an, sums = 4, {0, 2, 3, 4, 5, 6, 7, 9}
    yield from [0, 2, 3, 4]
    while True:
        an = next(k for k in count(an+1) if k not in sums)
        yield an
        sums |= {an+s for s in sums}
print(list(islice(agen(), 25))) # Michael S. Branicky, Nov 07 2025

Crossrefs

Cf. A003662, A003663, A005408, A026471, A026474, A033627, A051039, A051040, A244151, A244749.

Cf. A060469, A060470, A060471, A060472.

Cf. A000079.

Sequence in context: A339973 A224912 A162724 * A140974 A349763 A360000

Adjacent sequences: A244747 A244748 A244749 * A244751 A244752 A244753

Keyword

nonn

Author

N. J. A. Sloane and Robert G. Wilson v, Jul 05 2014

Extensions

Corrected by R. J. Mathar, Jul 12 2014

Status

approved