cp's OEIS Frontend

From Gus Wiseman, Aug 28 2023: (Start)

Appears to be the number of subsets of {1..n} containing n such that no element can be written as a nonnegative linear combination of the others, first differences of A326083. For example, the a(1) = 1 through a(8) = 10 subsets are:

{1} {2} {3} {4} {5} {6} {7} {8}

{2,3} {3,4} {2,5} {4,6} {2,7} {3,8}

{3,5} {5,6} {3,7} {5,8}

{4,5} {4,5,6} {4,7} {6,8}

{3,4,5} {5,7} {7,8}

{6,7} {3,7,8}

{3,5,7} {5,6,8}

{4,5,7} {5,7,8}

{4,6,7} {6,7,8}

{5,6,7} {5,6,7,8}

{4,5,6,7}

Note that these subsets do not all generate numerical semigroups, as their GCD is unrestricted, cf. A358392. The complement is counted by A365046, first differences of A364914.

(End)

From Johannes W. Meijer, May 29 2010: (Start)

a(n) is the number of ways White can force checkmate in exactly (n+1) moves, n >= 0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6, g5 and h6; Black Ke8, Nh8, pawns b3, c7, d3, f7, g6 and h7. (After Noam D. Elkies, see link; diagram 5).

Counts all paths of length n, n >= 0, starting at the third node on the path graph P_5, see the Maple program. (End)

From Alec Jones, Feb 25 2016: (Start)

The a(n) are the n-th terms in a "Fibonacci snake" drawn on a rectilinear grid. The n-th term is computed as the sum of the previous terms in cells adjacent to the n-th cell (diagonals included). (This sequence excludes the first term of the snake.)

For example:

1 ... 1 1 ... 1 4 1 4 6 ... 1 4 6 1 4 6 ... and so on.

1 ... 1 2 1 2 ... 1 2 1 2 12 ... 1 2 12 18 (End)

From Gus Wiseman, Oct 06 2023: (Start)

Also the number of subsets of {1..n} containing no two distinct elements summing to n. The a(0) = 1 through a(4) = 12 subsets are:

{} {} {} {} {}

{1} {1} {1} {1}

{2} {2} {2}

{1,2} {3} {3}

{1,3} {4}

{2,3} {1,2}

{1,4}

{2,3}

{2,4}

{3,4}

{1,2,4}

{2,3,4}

For n+1 instead of n we have A038754, complement A167762.

Including twins gives A117855, complement A366131.

The complement is counted by A365544.

For all subsets (not just pairs) we have A365377, complement A365376. (End)

Number of different values of A007947(m) when A056239(m) is equal to n.

From Gus Wiseman, Sep 11 2023: (Start)

Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:

{1} {1} {1} {1} {1} {1} {1}

{2} {3} {2} {5} {2} {7}

{1,2} {4} {1,2} {3} {1,2}

{1,2} {1,3} {6} {1,3}

{1,3} {1,4} {1,2} {1,4}

{2,3} {1,3} {1,5}

{1,4} {1,6}

{1,5} {2,3}

{2,4} {2,5}

{1,2,3} {3,4}

{1,2,3}

{1,2,4}

Cf. A365073, A365311, A365312, A365322, A365380.

(End)

Includes all subsets containing both 1 and n.

Rows are palindromic.

Using the standard definition of sum-free set, this is simply the difference of successive terms in A007865.

Number of subsets of {1..n} containing n but not containing the sum of any other two elements (repeats allowed). Also the number of sum-free sets (A007865) with maximum n. - Gus Wiseman, Aug 12 2023

Or, partitions whose length is a subset-sum of the parts.

These partitions have Heinz numbers A367224 /\ A005117.