A216999 Number of integers obtainable from 1 in n steps using addition, multiplication, and subtraction.

1, 3, 6, 13, 38, 153, 867, 6930, 75986, 1109442, 20693262, 477815647

Offset: 0

Author

Stan Wagon, Sep 22 2012

Keywords

• nonn
• more
• hard
• nice

Comments

A straight-line program is a sequence that starts at 1 and has each entry obtained from two preceding entries by addition, multiplication, or subtraction. S(n) is the set of integers obtainable at any point in a straight-line program using n steps. Thus S(0) = {1}, S(1) = {0,1,2}, S(2) = {-1,0,1,2,3,4}; the sequence here is the cardinality of S(n).

Links

• Peter Borwein and Joe Hobart, The extraordinary power of division in straight line programs, American Mathematical Monthly 119:7 (2012), pp. 584-592.
• Michael Shub and Steve Smale, On the intractability of Hilbert's Nullstellensatz and an algebraic version of "NP = P", Duke Mathematical Journal 81:1 (1995), pp. 47-54.

Crossrefs

Cf. A003065, A141414, A173419, A288849, A288850.

Programs

Mathematica

extend[p_] :=  Module[{q = Tuples[p, {2}], new},
  new = Flatten[Table[{Total[t], Subtract @@ t, Times @@ t}, {t, q}]];
  Union[ Sort /@  DeleteCases[ Table[If[! MemberQ[p, n], Append[p, n]], {n, new}], Null]]] ;
P[0] = {{1}};
P[n_] := P[n] = DeleteDuplicates[Flatten[extend /@ P[n - 1], 1]];
S[n_] := DeleteDuplicates[Flatten[P[n]]];
Length /@ S /@ Range[6]

Extensions

a(9)-a(11) (Michael Collier verified independently the 1109442, 20693262 values) by Gil Dogon, Sep 27 2013