OFFSET
0,3
COMMENTS
Let B = {0,1,2,3,6,10,15,21,...}, so that B consists of the triangular numbers together with 0 and 2. We call B the enhanced basis of triangular numbers. Define R(0) = 0 and R(n) = g(n) + R(n - g(n)) for n > 0, where g(n) is the greatest number in B that is <= n. Thus, each n has an enhanced triangular-number representation of the form R(n) = b(m(n)) + b(m(n-1)) + ... + b(m(k)), where b(n) > m(n-1) > ... > m(k) > 0, in which the last term, b(m(k)) is the trace.
The least n for which R(n) has more than 4 terms is given by R(7259) = 7140 + 105 + 10 + 3 + 1.
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..1000
EXAMPLE
R(0) = 0;
R(1) = 1;
R(2) = 2;
R(3) = 3;
R(4) = 3 + 1;
R(5) = 3 + 2;
R(6) = 6;
R(119) = 105 + 10 + 3 + 1.
MATHEMATICA
b[n_] := n (n + 1)/2; bb = Insert[Table[b[n], {n, 0, 200}], 2, 3]
s[n_] := Table[b[n], {k, 1, n + 1}];
h[1] = {0, 1, 2}; h[n_] := Join[h[n - 1], s[n]];
g = h[200]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
t = Table[r[n], {n, 0, 120}] (*A256909 before concatenation*)
Table[Last[r[n]], {n, 0, 120}] (*A256910*)
Table[Length[r[n]], {n, 0, 120}] (*A256911*)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 13 2015
STATUS
approved