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

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

Offset

1,3

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<a1<a2<...

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

Links

Table of n, a(n) for n=1..7.

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 *)

Crossrefs

Cf. A008932, A152111.

Sequence in context: A354436 A157451 A188204 * A167810 A331646 A054420

Adjacent sequences: A152109 A152110 A152111 * A152113 A152114 A152115

Keyword

nonn,more

Author

David S. Newman, Mar 22 2009

Extensions

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

Status

approved