A292997 - OEIS

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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

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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, 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.