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Trace of n in the minimal alternating squares representation of n.
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%I #8 Apr 14 2015 11:04:42

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

%T -9,1,-2,-1,4,-4,1,2,-1,36,4,-4,1,-9,1,-2,-1,4,-4,1,2,-1,49,-2,-1,4,

%U -4,1,-9,1,-2,-1,4,-4,1,2,-1,64,-16,1,-2,-1,4,-4,1,-9

%N Trace of n in the minimal alternating squares representation of n.

%C See A256789 for definitions.

%C For each positive integer m, the list of 2m numbers between m^2 and (m+1)^2 is repeated between (m+1)^2 and (m+2)^2. Consequently, a limiting sequence is formed by reversing the repeated lists. The limiting sequence is -1, 2, 1, -4, 4, -1, -2, 1, -9, 1, -4, 4, -1, -2, 1, -16, ...

%H Clark Kimberling, <a href="/A256791/b256791.txt">Table of n, a(n) for n = 0..999</a>

%e R(0) = 0, so a(0) = 0;

%e R(1) = 1, so a(1) = 1;

%e R(2) = 4 - 2, so a(2) = -2;

%e R(7) = 9 - 4 + 2, so a(7) = 2;

%e R(89) = 100 - 16 + 9 - 4, so a(89) = -4.

%t b[n_] := n^2; bb = Table[b[n], {n, 0, 1000}];

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

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

%t g = h[100]; r[0] = {0}; r[1] = {1}; r[2] = {4, -2};

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

%t Table[r[n], {n, 0, 120}] (* A256789 *)

%t Flatten[Table[Last[r[n]], {n, 0, 100}]] (* A256791 *)

%Y Cf. A256789.

%K easy,sign

%O 0,3

%A _Clark Kimberling_, Apr 13 2015