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A215022
NegaFibonacci representation code for n.
13
0, 1, 100, 101, 10010, 10000, 10001, 10100, 10101, 1001010, 1001000, 1001001, 1000010, 1000000, 1000001, 1000100, 1000101, 1010010, 1010000, 1010001, 1010100, 1010101, 100101010, 100101000, 100101001, 100100010, 100100000, 100100001, 100100100, 100100101, 100001010
OFFSET
0,3
COMMENTS
Let F_{-n} be the negative Fibonacci numbers (as defined in the first comment in A039834): F_{-1}=1, F_{-2}=-1, F_{-3}=2, F_{-4}=-3, F_{-5}=5, ..., F_{-n}=(-1)^(n-1)F_n.
Every integer has a unique representation as n = Sum_{k=1..r} c_k F_{-k} for some r, where the c_k are 0 or 1 and no two adjacent c's are 1.
Then a(n) is the concatenation c_r ... c_3 c_2 c_1.
REFERENCES
Donald E. Knuth, The Art of Computer Programming, Volume 4A, Combinatorial algorithms, Part 1, Addison-Wesley, 2011, pp. 168-171.
LINKS
M. W. Bunder, Zeckendorf representations using negative Fibonacci numbers, The Fibonacci Quarterly, Vol. 30, No. 2 (1992), pp. 111-115.
Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 1: Bitwise Tricks & Techniques; Binary Decision Diagrams, a pre-publication draft of section 7.1.3, 2009, pp. 36-39.
EXAMPLE
4 = 5 - 1 = F_{-5} + F_{-2}, so a(4) = 10010.
MATHEMATICA
ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]]; f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i]; a[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 10^(i - 1); k -= Fibonacci[-i]]; s]; Array[a, 100, 0] (* Amiram Eldar, Oct 15 2019 *)
PROG
(PARI) a(n)=if(n<2, return(n)); my(s=1, k=1, v); while(s<n, s+=fibonacci(k+=2)); v=binary(2^k/2); n-=fibonacci(k); forstep(i=k-2, 1, -1, if(abs(n-fibonacci(-i))<abs(n), n-=fibonacci(-i); v[#v+1-i]=1; i--)); subst(Pol(v), x, 10) \\ Charles R Greathouse IV, Aug 03 2012 [Caution: returns wrong values for some values of n > 15. Amiram Eldar, Oct 15 2019]
CROSSREFS
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Aug 03 2012
EXTENSIONS
a(16) inserted and 1 term added by Amiram Eldar, Oct 11 2019
STATUS
approved