A102855 Minimal number of distinct nonzero tetrahedral numbers needed to represent n, or -1 if no such representation is possible.

1, -1, -1, 1, 2, -1, -1, -1, -1, 1, 2, -1, -1, 2, 3, -1, -1, -1, -1, 1, 2, -1, -1, 2, 3, -1, -1, -1, -1, 2, 3, -1, -1, 3, 1, 2, -1, -1, 2, 3, -1, -1, -1, -1, 2, 3, -1, -1, 3, 4, -1, -1, -1, -1, 2, 1, 2, -1, 3, 2, 3, -1, -1, -1, 3, 2, 3, -1, 4, 3, 4, -1, -1, -1, -1, 2, 3, -1, -1

Offset: 1

Author

Jud McCranie, Mar 01 2005

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A104246, A102795-A102806, A102856-A102858.

Programs

Maple

N:= 100; # for a(1)..a(N)
ft:= t -> t*(t+1)*(t+2)/6:
tets:= map(ft, [$1..floor((6*N)^(1/3))]:
f:= proc(n,tmax) option remember;
   local res, s;
   if member(n, tets) and n < tmax then return 1 fi;
   min(seq(1 + procname(n-s,s), s=select(`<`,tets,min(n,tmax))));
end proc:
subs(infinity=-1,map(f, [$1..N],infinity)); # Robert Israel, Dec 29 2019

Mathematica

M = 100; (* number of terms *)
ft[t_] := t(t+1)(t+2)/6;
tets = ft /@ Range[1, Floor[(6M)^(1/3)]];
f[n_, tmax_] := f[n, tmax] = If[MemberQ[tets, n] && n < tmax, 1, Min[ Table[1 + f[n-s, s], {s, Select[tets, # < Min[n, tmax]&]}]]];
f[#, Infinity]& /@ Range[1, M] /. Infinity -> -1 (* Jean-François Alcover, Aug 05 2022, after Robert Israel *)