A242921 -id:A242921 - cp's OEIS frontend

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

Jeffrey Shallit, May 26 2014

nonn

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

First difference of A242921.

Cf. A191818.

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

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 *)

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.

A242924 Position of first occurrence of n in A242923.

Original entry on oeis.org

1, 2, 4, 8, 12, 66, 24, 34, 233, 251, 284, 173, 104, 299, 329, 431, 596, 625, 528, 1052, 759, 349, 667, 1028, 793, 436, 1242, 1882, 1410, 1374, 4974, 1181, 3626, 3517, 3673, 3148, 4398, 6160, 5537, 4254, 5512, 7039, 4074, 2194, 10206, 11361, 4154, 12710, 7559

Offset: 1

Jeffrey Shallit, May 26 2014

nonn

Not currently known to be finite for all n.

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A242921, A242923.

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:= proc() local t, a; t, a:= 0, proc() 0 end;
      proc(n) local h;
        while a(n) = 0 do
          t:= t+1; h:= b(t) -b(t-1);
          if a(h) = 0 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=1..30);  # Alois P. Heinz, May 26 2014

nmaxb = 2000; (* max index of b(n) *)
nmaxa = 30; (* max index of a(n) *)
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 - 2 i] + t != 2 b[n - i]]; If[ok, Return[t]]]]];
B = Array[b, nmaxb] // Differences;
a[n_] := a[n] = Module[{p = FirstPosition[B, n]}, Which[n == 1, 1, p === Missing["NotFound"], -1, True, p[[1]] + 1]];
Array[a, nmaxa] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz for b(n) *)

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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.

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