A256795 Difference sequence of A256793.

1, 2, 3, 2, 2, 3, 3, 1, 2, 3, 3, 2, 1, 2, 3, 3, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 3, 3, 1, 2, 1, 2, 2, 2, 1, 2, 3, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 3, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 3, 3, 1, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 3, 2, 1

Offset: 1

Author

Clark Kimberling, Apr 13 2015

Keywords

• nonn
• easy

Comments

These are the numbers of consecutive positive traces when the minimal alternating squares representations for positive integers are written in order. Is every term < 5? The first term greater than 3 is a(116) = 4, corresponding to these 3 consecutive representations:

R(225) = 225;

R(226) = 256 - 36 + 9 - 4 + 1;

R(227) = 256 - 36 + 9 - 4 + 2.

(See A256789 for definitions.)

Crossrefs

Cf. A256792, A256794.

Programs

Mathematica

b[n_] := n^2; bb = Table[b[n], {n, 0, 1000}];  (* Squares as base *)
s[n_] := Table[b[n], {k, 1, 2 n - 1}];
h[1] = {1}; h[n_] := Join[h[n - 1], s[n]];
g = h[100]; r[0] = {0}; r[1] = {1}; r[2] = {4, -2};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
Table[r[n], {n, 0, 120}]; (* A256789 *)
u = Flatten[Table[Last[r[n]], {n, 1, 1000}]];  (* A256791 *)
u1 = Select[Range[800], u[[#]] > 0 &]; (* A256792 *)
u2 = Select[Range[800], u[[#]] < 0 &]; (* A256793 *)
Differences[u1]  (* A256794 *)
Differences[u2]  (* A256795 *)