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Positive part of the minimal alternating triangular-number representation of n (defined at A255974).
2

%I #4 Apr 13 2015 09:40:13

%S 1,3,3,7,6,6,10,11,10,10,18,15,16,15,15,22,24,21,22,21,21,28,29,31,28,

%T 29,28,28,39,36,37,39,36,37,36,36,48,48,45,46,48,45,46,45,45,56,58,58,

%U 55,56,58,55,56,55,55,66,67,69,69,66,67,69,66,67,66,66

%N Positive part of the minimal alternating triangular-number representation of n (defined at A255974).

%H Clark Kimberling, <a href="/A256700/b256700.txt">Table of n, a(n) for n = 1..1000</a>

%F A256700(n) - A256759(n) = n.

%e R(1) = 1; positive part 1, nonpositive part 0

%e R(2) = 3 - 1; positive part 3, nonpositive part 1

%e R(3) = 3; positive part 3, nonpositive part 0

%e R(11) = 15 - 6 + 3 - 1; positive part 15+3 = 18; nonpositive part 6 + 1 = 7

%t b[n_] := n (n + 1)/2; bb = Table[b[n], {n, 0, 1000}];

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

%t h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[100]; r[0] = {0};

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

%t Table[Total[(Abs[r[n]] + r[n])/2], {n, 1, 120}] (* A256700 *)

%t Table[Total[(Abs[r[n]] - r[n])/2], {n, 1, 120}] (* A256759 *)

%Y Cf. A255974, A256759.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Apr 11 2015