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
Offset: 1
Examples
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.
References
- R. K. Guy, "s-Additive sequences," preprint, 1994.
Links
- Steven R. Finch, Are 0-additive sequences always regular?, Amer. Math. Monthly, 99 (1992), 671-673.
Crossrefs
Programs
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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
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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]
Extensions
Corrected by R. J. Mathar, Jul 12 2014