cp's OEIS Frontend

A152112 Number of increasing initial sequences of bases of order 3.

%I A152112 #15 Feb 24 2023 11:25:23
%S A152112 1,1,3,13,86,760,8518
%N A152112 Number of increasing initial sequences of bases of order 3.
%C A152112 Using the terminology of A008932, call a set A a basis of order h if every number can be written as the sum of h (not necessarily distinct) elements of A. Call a basis an increasing basis of order h if its elements are arranged in increasing order, a0<a1<a2<...
%C A152112 Consider the set of all initial subsequences of any length {a0, a1, a2, ..., an} of all the increasing bases. These can be ordered in lexicographic order, giving, for h = 3:
%C A152112 0
%C A152112 0,1
%C A152112 0,1,2
%C A152112 0,1,3
%C A152112 0,1,4
%t A152112 f[A_]:=
%t A152112 (AAA={};
%t A152112 For [ii=1,ii<=Length[A],ii++,
%t A152112 For[jj=1,jj<=Length[A],jj++,
%t A152112 For [kk=1,kk<=Length[A],kk++,
%t A152112 AAA=Union[AAA,{A[[ii]]+A[[jj]]+A[[kk]]}]]]];
%t A152112 For[ii=1,ii<=Length[AAA],ii++,
%t A152112 If[ii==Length[AAA],max=ii-1];
%t A152112 If[AAA[[ii]]>ii-1,max=ii-2;Break[]]]);
%t A152112 index=1;
%t A152112 seq[1]={0,1};
%t A152112 rindex=1;
%t A152112 newindex=1;
%t A152112 For[k=1,k<=5,k++,
%t A152112 jbegin=rindex;jend=index;
%t A152112 For[j=jbegin,j<=jend,j++,
%t A152112 f[seq[j]];
%t A152112 For[i=Max[seq[j]]+1,i<=max+1,i++,index++;seq[index]=Append[seq[rindex],i]
%t A152112 ];rindex=rindex+1;
%t A152112 ]]
%t A152112 For[i=1,i<=index,i++,Print[i," ",seq[i]]] (* _David S. Newman_, Dec 29 2014 *)
%Y A152112 Cf. A008932, A152111.
%K A152112 nonn,more
%O A152112 1,3
%A A152112 _David S. Newman_, Mar 22 2009
%E A152112 a(6)-a(7) from _David S. Newman_, Dec 29 2014