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 A256655 R(k), the minimal alternating Fibonacci representation of k, concatenated for k = 0, 1, 2,.... 15
 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, -8, 2, 21, -5, 21, -5, 1, 21, -3, 21, -2, 21, -1, 21, 34, -13, 1, 34, -13, 2, 34, -13, 3, 34, -13, 5, -1, 34, -8, 34, -8, 1, 34, -8, 2, 34, -5, 34, -5, 1, 34, -3, 34 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. If b(n) = F(n+2), where F = A000045, then the sum is the minimal alternating Fibonacci-representation of n. LINKS Clark Kimberling, Table of n, a(n) for n = 0..1000 FORMULA R(F(k)^2) = F(2k-1) - F(2k-3) + F(2k-5) - ... + d*F(5) + (-1)^k, where d = (-1)^(k+1). EXAMPLE R(0) = 0 R(1) = 1 R(2) = 2 R(3) = 3 R(4) = 5 - 1 R(9) = 13 - 5 + 1 R(25) = 34 - 13 + 5 - 1 R(64) = 89 - 34 + 13 - 5 + 1 MATHEMATICA f[n_] = Fibonacci[n]; ff = Table[f[n], {n, 1, 70}]; s[n_] := Table[f[n + 2], {k, 1, f[n]}]; h[0] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[12]; r[0] = {0}; r[n_] := If[MemberQ[ff, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]; Flatten[Table[r[n], {n, 0, 60}]] CROSSREFS 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). Sequence in context: A039704 A002752 A239693 * A128047 A105870 A096534 Adjacent sequences:  A256652 A256653 A256654 * A256656 A256657 A256658 KEYWORD easy,sign AUTHOR Clark Kimberling, Apr 08 2015 STATUS approved

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Last modified December 19 08:25 EST 2018. Contains 318245 sequences. (Running on oeis4.)