A292997 Irregular triangle read by rows: lexicographically earliest, minimum size sets, S_n, such that {1, 2, ... n} is a subset of the union of S_n with the sums of distinct pairs in S_n and the differences of pairs in S_n.

1, 1, 2, 1, 2, 1, 3, 1, 2, 3, 1, 2, 4, 1, 2, 5, 1, 2, 6, 1, 2, 3, 6, 1, 2, 3, 7, 1, 2, 3, 8, 1, 2, 3, 9, 1, 2, 6, 11, 1, 2, 3, 4, 10, 1, 2, 3, 4, 11, 1, 2, 3, 4, 12, 1, 2, 3, 7, 14, 1, 2, 3, 8, 15, 1, 2, 3, 9, 16, 1, 2, 7, 13, 17, 1, 2, 3, 4, 8, 17

Offset

1,3

Comments

Each row is weakly longer than the preceding row, and weakly lexicographically later if the preceding row is the same length.

Conjecture: A225687 gives the row lengths.

Links

Peter Kagey, Table of n, a(n) for n = 1..265 (first 47 rows, flattened)

Reddit user uoaei, For a given maximum integer N, what is the set of integers with minimal cardinality that can give me any integer between 1 and N by only one addition or subtraction between any two elements in that set?

Example

For n = 7, S_7 = {1, 2, 5}. Because {1, 2, ..., 7} can be represented as {1, 2, 1 + 2, 5 - 1, 5, 5 + 1, 5 + 2}, where each term is in S_7 or is the sum or difference of two distinct terms in S_7.
There is no set of fewer than three elements with this property, and S_7 is the lexicographically earliest set of three elements with this property.
Table begins:
  1;
  1, 2;
  1, 2;
  1, 3;
  1, 2, 3;
  1, 2, 4;
  1, 2, 5;
  1, 2, 6;
  1, 2, 3, 6;
  1, 2, 3, 7.

Crossrefs

Cf. A005228, A225687.

Sequence in context: A029436 A263274 A333940 * A372471 A060135 A057112

Adjacent sequences: A292994 A292995 A292996 * A292998 A292999 A293000

Keyword

nonn,tabf

Author

Peter Kagey, Sep 27 2017

Status

approved