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A048249 Number of distinct values produced from sums and products of n unity arguments. 12
1, 2, 3, 4, 6, 9, 11, 17, 23, 30, 44, 60, 80, 114, 156, 212, 296, 404, 556, 770, 1065, 1463, 2032, 2795, 3889, 5364, 7422, 10300, 14229, 19722, 27391, 37892, 52599, 73075, 101301, 140588, 195405, 271024, 376608, 523518, 726812, 1010576 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Values listed calculated by exhaustive search algorithm.

For n+1 operands (n operations) there are (2n)!/((n!)((n+1)!)) possible postfix forms over a single operator. For each such form, there are 2^n ways to assign 2 operators (here, sum and product). Calculate results and eliminate duplicates.

Number of distinct positive integers that can be obtained by iteratively adding or multiplying together parts of an integer partition until only one part remains, starting with 1^n. - Gus Wiseman, Sep 29 2018

LINKS

Table of n, a(n) for n=1..42.

Index to sequences related to the complexity of n

Index entries for similar sequences

FORMULA

Equals partial sum of "number of numbers of complexity n" (A005421). - Jonathan Vos Post, Apr 07 2006

EXAMPLE

a(3)=3 since (in postfix): 111** = 11*1* = 1, 111*+ = 11*1+ = 111+* = 11+1* = 2 and 111++ = 11+1+ = 3. Note that at n=7, the 11 possible values produced are the set {1,2,3,4,5,6,7,8,9,10,12}. This is the first n for which there are "skipped" values in the set.

MATHEMATICA

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];

Table[Length[Select[ReplaceListRepeated[{Array[1&, n]}, {{foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x+y]], {foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}], Length[#]==1&]], {n, 10}] (* Gus Wiseman, Sep 29 2018 *)

CROSSREFS

Cf. A000792, A005520, A066739, A070960, A201163, A319850, A318949, A319855, A319856, A319909, A319910, A319911.

Sequence in context: A007210 A198394 A035947 * A288734 A318759 A018471

Adjacent sequences:  A048246 A048247 A048248 * A048250 A048251 A048252

KEYWORD

nonn,nice

AUTHOR

Tony Bartoletti

EXTENSIONS

More terms from David W. Wilson, Oct 10 2001

STATUS

approved

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Last modified January 18 23:05 EST 2019. Contains 319282 sequences. (Running on oeis4.)