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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.

%I A242923 #18 May 01 2022 07:23:15
%S A242923 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,
%T A242923 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,
%U A242923 3,6,1,6,2,3,2,1,2,8,3,1,2,1,3,1,2,7,1
%N 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.
%H A242923 Alois P. Heinz, <a href="/A242923/b242923.txt">Table of n, a(n) for n = 1..10000</a>
%F A242923 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.
%p A242923 b:= proc(n) option remember; local i, t, ok;
%p A242923       if n<2 then n
%p A242923     else for t from 1+b(n-1) do ok:=true;
%p A242923            for i to n/2 while ok
%p A242923              do ok:=b(n-2*i)+t <> 2*b(n-i) od;
%p A242923            if ok then return t fi
%p A242923          od
%p A242923       fi
%p A242923     end:
%p A242923 a:= n-> b(n) -b(n-1):
%p A242923 seq(a(n), n=1..120);  # _Alois P. Heinz_, May 26 2014
%t A242923 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]]]]];
%t A242923 a[n_] := b[n] - b[n-1];
%t A242923 Array[a, 120] (* _Jean-François Alcover_, Nov 13 2020, after _Alois P. Heinz_ *)
%Y A242923 First difference of A242921.
%Y A242923 Cf. A191818.
%K A242923 nonn
%O A242923 1,2
%A A242923 _Jeffrey Shallit_, May 26 2014