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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 August 17 18:38 EDT 2017. Contains 290655 sequences.