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A256789
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R(k), the minimal alternating squares representation of k, concatenated for k = 0, 1, 2,....
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13
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0, 1, 4, -2, 4, -1, 4, 9, -4, 9, -4, 1, 9, -4, 2, 9, -1, 9, 16, -9, 4, -1, 16, -9, 4, 16, -4, 16, -4, 1, 16, -4, 2, 16, -1, 16, 25, -9, 1, 25, -9, 4, -2, 25, -9, 4, -1, 25, -9, 4, 25, -4, 25, -4, 1, 25, -4, 2, 25, -1, 25, 36, -16, 9, -4, 1, 36, -9, 36, -9, 1
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OFFSET
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0,3
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COMMENTS
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Let B(n) be the least square >= n. The minimal alternating squares representation of a nonnegative integer n is defined as the sum B(n) - B(m(1) + B(m(2) + ... + d*B(m(k)) that results from the recurrence R(n) = B(n) - R(B(n) - n), with initial representations R(0) = 0, R(1) = 1, and R(2) = 4 - 2. The sum B(n) + B(m(2)) + ... is the positive part of R(n), and the sum B(m(1)) + B(m(3)) + ... is the nonpositive part of R(n). The last term of R(k) is the trace of n. If b(n) = n(n+1)/2, the n-th triangular number, then the sum R(n) is the minimal alternating triangular-number representation of n.
Unlike minimal alternating representations for other bases (e.g., Fibonacci numbers, A256655; binary, A256696, triangular numbers, A244974), the trace of a minimal alternating squares representation is not necessarily a member of the base; specifically, the trace can be -2 or 2, which are not squares.
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LINKS
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EXAMPLE
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R(0) = 0
R(1) = 1
R(2) = 4 - 2
R(3) = 4 - 1
R(4) = 4
R(5) = 9 - 4
R(6) = 9 - 4 + 1
R(7) = 9 - 4 + 2
R(89) = 100 - 16 + 9 - 4
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MATHEMATICA
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b[n_] := n^2; bb = Table[b[n], {n, 0, 1000}];
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, individual representations *)
Flatten[Table[r[n], {n, 0, 120}]] (* A256789, concatenated representations *)
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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