A256910 - OEIS

%I #6 Apr 14 2015 11:05:48

%S 0,1,2,3,1,2,6,1,2,3,10,1,2,3,1,15,1,2,3,1,2,21,1,2,3,1,2,6,28,1,2,3,

%T 1,2,6,1,36,1,2,3,1,2,6,1,2,45,1,2,3,1,2,6,1,2,3,55,1,2,3,1,2,6,1,2,3,

%U 10,66,1,2,3,1,2,6,1,2,3,10,1,78,1,2,3,1

%N Trace of the enhanced triangular-number representation of n.

%C See A256909 for definitions.

%H Clark Kimberling, <a href="/A256910/b256910.txt">Table of n, a(n) for n = 0..1000</a>

%e R(0) = 0, trace = 0;

%e R(1) = 1, trace = 1;

%e R(2) = 2, trace = 2;

%e R(3) = 3, trace = 3;

%e R(4) = 3 + 1, trace = 1;

%e R(5) = 3 + 2, trace = 2;

%e R(6) = 6, trace = 6;

%e R(119) = 105 + 10 + 3 + 1, trace = 1.

%t b[n_] := n (n + 1)/2; bb = Insert[Table[b[n], {n, 0, 200}], 2, 3]

%t s[n_] := Table[b[n], {k, 1, n + 1}];

%t h[1] = {0, 1, 2}; h[n_] := Join[h[n - 1], s[n]];

%t g = h[200]; r[0] = {0};

%t r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];

%t t = Table[r[n], {n, 0, 120}] (*A256909 before concatenation*)

%t Flatten[t]  (*A256909*)

%t Table[Last[r[n]], {n, 0, 120}]    (*A256910*)

%t Table[Length[r[n]], {n, 0, 120}]  (*A256911*)

%Y Cf. A000217, A256909 (definitions), A256911 (number of terms), A255974 (minimal alternating triangular-number representations).

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Apr 13 2015