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A257022
Trace of n in the quarter-sum representation of n.
4
0, 1, 2, 1, 4, 1, 6, 1, 2, 9, 1, 2, 12, 1, 2, 1, 16, 1, 2, 1, 20, 1, 2, 1, 4, 25, 1, 2, 1, 4, 30, 1, 2, 1, 4, 1, 36, 1, 2, 1, 4, 1, 42, 1, 2, 1, 4, 1, 6, 49, 1, 2, 1, 4, 1, 6, 56, 1, 2, 1, 4, 1, 6, 1, 64, 1, 2, 1, 4, 1, 6, 1, 72, 1, 2, 1, 4, 1, 6, 1, 2, 81
OFFSET
0,3
COMMENTS
Every positive integer is a sum of at most four distinct quarter squares, of which the least term is the trace; see A257019.
LINKS
EXAMPLE
Quarter-square representations:
r(0) = 0, so a(0) = 0
r(1) = 1, so a(1) = 1
r(2) = 2, so a(2) = 2
r(3) = 2 + 1, so a(3) = 1
MATHEMATICA
z = 100; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, 100}];
s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
h[1] = {1}; h[n_] := Join[h[n - 1], s[n]];
g = h[200]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
Table[Last[r[n]], {n, 0, 3 z}] (* A257022 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 15 2015
STATUS
approved