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A256655 R(k), the minimal alternating Fibonacci representation of k, concatenated for k = 0, 1, 2,....

%I A256655 #7 Apr 13 2015 09:38:57
%S A256655 0,1,2,3,5,-1,5,8,-2,8,-1,8,13,-5,1,13,-3,13,-2,13,-1,13,21,-8,1,21,
%T A256655 -8,2,21,-5,21,-5,1,21,-3,21,-2,21,-1,21,34,-13,1,34,-13,2,34,-13,3,
%U A256655 34,-13,5,-1,34,-8,34,-8,1,34,-8,2,34,-5,34,-5,1,34,-3,34
%N A256655 R(k), the minimal alternating Fibonacci representation of k, concatenated for k = 0, 1, 2,....
%C A256655 Suppose that b = (b(0), b(1), ... ) is an increasing sequence of positive integers satisfying b(0) = 1 and b(n+1) <= 2*b(n) for n >= 0.  Let B(n) be the least b(m) >= n.  Let R(0) = 1, and for n > 0, let R(n) = B(n) - R(B(n) - n).  The resulting sum of the form R(n) = B(n) - B(m(1)) + B(m(2)) - ... + ((-1)^k)*B(k) is introduced here as the minimal alternating b-representation of n.  The sum B(n) + B(m(2)) + ... we call the positive part of R(n), and the sum B(m(1)) + B(m(3)) + ... , the nonpositive part of R(n).  The number ((-1)^k)*B(k) is the trace of n.
%C A256655 If b(n) = F(n+2), where F = A000045, then the sum is the minimal alternating Fibonacci-representation of n.
%H A256655 Clark Kimberling, <a href="/A256655/b256655.txt">Table of n, a(n) for n = 0..1000</a>
%F A256655 R(F(k)^2) = F(2k-1) - F(2k-3) + F(2k-5) - ... + d*F(5) + (-1)^k, where d = (-1)^(k+1).
%e A256655 R(0) = 0
%e A256655 R(1) = 1
%e A256655 R(2) = 2
%e A256655 R(3) = 3
%e A256655 R(4) = 5 - 1
%e A256655 R(9) = 13 - 5 + 1
%e A256655 R(25) = 34 - 13 + 5 - 1
%e A256655 R(64) = 89 - 34 + 13 - 5 + 1
%t A256655 f[n_] = Fibonacci[n]; ff = Table[f[n], {n, 1, 70}];
%t A256655 s[n_] := Table[f[n + 2], {k, 1, f[n]}];
%t A256655 h[0] = {1}; h[n_] := Join[h[n - 1], s[n]];
%t A256655 g = h[12]; r[0] = {0};
%t A256655 r[n_] := If[MemberQ[ff, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
%t A256655 Flatten[Table[r[n], {n, 0, 60}]]
%Y A256655 Cf. A000045, A255973 (trace), A256656 (numbers with positive trace), A256657 (numbers with nonpositive trace), A256663 (positive part of R(n)), A256664 (nonpositive part of R(n)), A256654, A256696 (minimal alternating binary representations), A255974 (minimal alternating triangular-number representations), A256789 (minimal alternating squares representations).
%K A256655 easy,sign
%O A256655 0,3
%A A256655 _Clark Kimberling_, Apr 08 2015