A242923 Lexicographically least sequence over the positive integers with the property that there are no two consecutive blocks of the same length and same sum.

1, 2, 1, 3, 1, 2, 1, 4, 2, 1, 2, 5, 2, 1, 3, 1, 2, 1, 3, 4, 1, 2, 1, 7, 2, 3, 1, 2, 1, 5, 1, 2, 1, 8, 2, 4, 2, 3, 2, 1, 5, 4, 3, 7, 2, 1, 4, 2, 5, 3, 1, 2, 1, 3, 4, 1, 2, 1, 4, 5, 3, 2, 1, 7, 4, 6, 2, 6, 3, 6, 1, 6, 2, 3, 2, 1, 2, 8, 3, 1, 2, 1, 3, 1, 2, 7, 1

Offset: 1

Author

Jeffrey Shallit, May 26 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

First difference of A242921.

Cf. A191818.

Programs

Maple

b:= proc(n) option remember; local i, t, ok;
      if n<2 then n
    else for t from 1+b(n-1) do ok:=true;
           for i to n/2 while ok
             do ok:=b(n-2*i)+t <> 2*b(n-i) od;
           if ok then return t fi
         od
      fi
    end:
a:= n-> b(n) -b(n-1):
seq(a(n), n=1..120);  # Alois P. Heinz, May 26 2014

Mathematica

b[n_] := b[n] = Module[{i, t, ok}, If[n<2, n, For[t = 1+b[n-1], True, t++, ok = True; For[i = 1, i <= n/2 && ok, i++, ok = b[n-2i] + t != 2b[n-i]]; If[ok, Return[t]]]]];
a[n_] := b[n] - b[n-1];
Array[a, 120] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)

Formula

a(9) = 2 because choosing a(9) = 1 gives the blocks (3,1,2) and (1,4,1), which are both of length 3 and sum to 6.