A256914 - OEIS

%I #10 Apr 15 2015 13:30:56

%S 0,1,2,3,4,1,2,3,1,9,1,2,3,4,1,2,16,1,2,3,4,1,2,3,1,25,1,2,3,4,1,2,3,

%T 1,9,1,36,1,2,3,4,1,2,3,1,9,1,2,3,49,1,2,3,4,1,2,3,1,9,1,2,3,4,1,64,1,

%U 2,3,4,1,2,3,1,9,1,2,3,4,1,2,16,81,1,2

%N Trace of the enhanced squares representation of n.

%C See A256913 for definitions.

%C a(A257046(n)) = 1; a(A257047(n)) != 1. - _Reinhard Zumkeller_, Apr 15 2015

%H Clark Kimberling, <a href="/A256914/b256914.txt">Table of n, a(n) for n = 0..1000</a>

%e R(0) = 0, so trace = 0.

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

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

%e R(43) = 36 + 4 + 3, so trace = 3.

%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 a256914 = last . a256913_row  -- _Reinhard Zumkeller_, Apr 15 2015

%Y Cf. A000290, A256913, A256915 (number of terms).

%Y Cf. A257046, A257047, A257070.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Apr 14 2015