%I #8 Apr 15 2015 13:30:43

%S 1,1,1,1,1,2,2,2,3,1,2,2,2,2,3,3,1,2,2,2,2,3,3,3,4,1,2,2,2,2,3,3,3,4,

%T 2,3,1,2,2,2,2,3,3,3,4,2,3,3,3,1,2,2,2,2,3,3,3,4,2,3,3,3,3,4,1,2,2,2,

%U 2,3,3,3,4,2,3,3,3,3,4,4,2,1,2,2,2,2

%N Length of the enhanced squares representation of n.

%C See A256913 for definitions.

%H Clark Kimberling, <a href="/A256915/b256915.txt">Table of n, a(n) for n = 0..1000</a>

%e R(0) = 0, so length = 1.

%e R(1) = 1, so length = 1.

%e R(8) = 4 + 3 + 1, so length = 3.

%e R(7224) = 7056 + 144 + 16 + 4 + 3 + 1, so length = 6.

%t b[n_] := n^2; bb = Insert[Table[b[n], {n, 0, 100}] , 2, 3];

%t s[n_] := Table[b[n], {k, 1, 2 n + 1}];

%t h[1] = {0, 1, 2, 3}; h[n_] := Join[h[n - 1], s[n]];

%t g = h[100]; Take[g, 100]

%t r[0] = {0}; r[1] = {1}; r[2] = {2}; r[3] = {3}; r[8] = {4, 3, 1};

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

%t t = Table[r[n], {n, 0, 120}] (* A256913, before concatenation *)

%t Flatten[t] (* A256913 *)

%t Table[Last[r[n]], {n, 0, 120}] (* A256914 *)

%t Table[Length[r[n]], {n, 0, 200}] (* A256915 *)

%o (Haskell)

%o a256915 = length . a256913_row -- _Reinhard Zumkeller_, Apr 15 2015

%Y Cf. A000290, A256913, A256914 (trace).

%Y Cf. A257071.

%K nonn,easy

%O 0,6

%A _Clark Kimberling_, Apr 14 2015