A229673 The number of subsets of nonzero integers of cardinality n, produced as the steps in a computation starting with 1 and using the operations of multiplication, addition, or subtraction.

1, 3, 15, 126, 1667, 31966, 828678, 27535826, 1128945382, 55473589278, 3193471420236, 211517309562652

Offset

1,2

Comments

A straight-line program (SLP) is a sequence that starts at 1 and has each entry obtained from two preceding entries by addition, multiplication, or subtraction. The length of the SLP is defined as that of the sequence without the first 1. An SLP is said to be reduced if there is no repetition in the sequence. Two SLPs are considered equivalent if their sequences define the same subset(only difference is sequence order). This sequence gives the number of SLPs with n steps in which 0 does not appear.

This sequence can also be thought of as defining the size of the search space that needs to be traversed when trying to compute other SLP related OEIS sequences such as A216999 or A229686. This is because 0 is never needed in the shortest SLP calculation of any other integer.

Michael Collier generated independently the values up to 1128945382.

Links

Table of n, a(n) for n=1..12.

Formula

a(n) >= 2^(n-1)*(n-1)!.

Example

a(1) = 1 and the SLP is (1,2).
a(2) = 3 and the SLPs are (1,2,3), (1,2,4), and (1,2,-1).

Crossrefs

Cf. A229662, A216999, A173419, A214872, A229686.

Sequence in context: A246573 A270503 A212553 * A266091 A135255 A182489

Adjacent sequences: A229670 A229671 A229672 * A229674 A229675 A229676

Keyword

nonn,hard

Author

Gil Dogon, Sep 27 2013

Status

approved