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

1, 1, 3, 13, 86, 760, 8518

Offset: 1

Author

David S. Newman, Mar 22 2009

Keywords

• nonn
• more

Comments

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

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:

0

0,1

0,1,2

0,1,3

0,1,4

Crossrefs

Cf. A008932, A152111.

Programs

Mathematica

f[A_]:=
(AAA={};
For [ii=1,ii<=Length[A],ii++,
For[jj=1,jj<=Length[A],jj++,
For [kk=1,kk<=Length[A],kk++,
AAA=Union[AAA,{A[[ii]]+A[[jj]]+A[[kk]]}]]]];
For[ii=1,ii<=Length[AAA],ii++,
If[ii==Length[AAA],max=ii-1];
If[AAA[[ii]]>ii-1,max=ii-2;Break[]]]);
index=1;
seq[1]={0,1};
rindex=1;
newindex=1;
For[k=1,k<=5,k++,
jbegin=rindex;jend=index;
For[j=jbegin,j<=jend,j++,
f[seq[j]];
For[i=Max[seq[j]]+1,i<=max+1,i++,index++;seq[index]=Append[seq[rindex],i]
];rindex=rindex+1;
]]
For[i=1,i<=index,i++,Print[i," ",seq[i]]] (* David S. Newman, Dec 29 2014 *)

Extensions

a(6)-a(7) from David S. Newman, Dec 29 2014

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