A380894 a(1) = 0; a(2) = 1; for n > 2, a(n) = a(n-1) + least unique positive difference of two earlier terms.

0, 1, 2, 4, 7, 11, 16, 22, 32, 44, 58, 76, 96, 121, 147, 177, 208, 241, 277, 314, 352, 392, 435, 480, 527, 579, 636, 694, 754, 815, 878, 943, 1014, 1086, 1159, 1235, 1312, 1390, 1470, 1551, 1634, 1719, 1806, 1894, 1989, 2085, 2185, 2286, 2389, 2494, 2600, 2709

Offset

1,3

Comments

For a(1) = k and a(2) = k + 1 the sequence is shifted by +k.

Links

Felix Huber, Table of n, a(n) for n = 1..1000

Formula

For n > 2, a(n) < (a(n+1) - a(n-1))/2.

Example

a(4) = 4 because a(3) = 2 and the difference 2 - 0 = 2 is unique and the difference 2 - 1 = 1 - 0 = 1 is not unique.
a(9) = 32 because a(8) = 22 and the difference 11 - 1 = 10 is unique, the differences 1, 2, 3, 4, 5, 6, 7, 9 are not unique and the difference 8 does not exist.

Maple

A380894:=proc(N)
    local i, j, L, M;
    M:=[0, 1];
    L:=[M[2]-M[1]];
    i:=2;
    while i<=N-1 do
        for j to i-2 do
            L:=[op(L), M[i]-M[j]];
        od;
        L:=sort(L);
        while numboccur(L[1], L)>1 do
            L:=remove(x->x=L[1], L);
        od;
        M:=[op(M), M[i]+L[1]];
        L:=subsop(1=NULL, L);
        i:=i+1;
    od;
    return op(M)
end proc;
A380894(52);

Mathematica

a[1] = 0; a[2] = 1; a[n_Integer] := a[n] = Module[{data = Array[a, n - 1, 1], len}, len = Length[data];  a[n - 1] + MinimalBy[DeleteCases[Tally[ResourceFunction["FlatTable"][data[[i]] - data[[j]], {j, 1, len - 1}, {i, j + 1, len}]], _?(#[[2]] > 1 &)], First][[1, 1]]]; Array[a, 52, 1] (* Shenghui Yang, Feb 18 2025 *)

Crossrefs

Cf. A000045 (Fibonacci numbers), A002858 (Ulam numbers with a(1) = 1 and a(2) = 2).

Sequence in context: A212365 A131075 A365698 * A133523 A114805 A196722

Adjacent sequences: A380891 A380892 A380893 * A380895 A380896 A380897

Keyword

nonn

Author

Felix Huber, Feb 10 2025

Status

approved