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A286298 a(0) = 0, a(1) = 1; thereafter, a(2n) = a(n) + 1 + (n mod 2), a(2n+1) = a(n) + 2 - (n mod 2). 2
0, 1, 3, 2, 4, 5, 4, 3, 5, 6, 7, 6, 5, 6, 5, 4, 6, 7, 8, 7, 8, 9, 8, 7, 6, 7, 8, 7, 6, 7, 6, 5, 7, 8, 9, 8, 9, 10, 9, 8, 9, 10, 11, 10, 9, 10, 9, 8, 7, 8, 9, 8, 9, 10, 9, 8, 7, 8, 9, 8, 7, 8, 7, 6, 8, 9, 10, 9, 10, 11, 10, 9, 10, 11, 12, 11, 10, 11, 10, 9, 10, 11, 12, 11, 12, 13, 12, 11, 10, 11, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let S be a set containing 0 and let S grow in generations G(i), defined by these rules: If x is in S then 2x is in S and 1 - x is in S. So G(0) = 0, G(1) = {1}, G(2) = {2}, G(3) = {-1,4}, ... Then a(n) is the generation containing the integer k where n = 2k - 1 for k>0 and -2k for k <= 0. The question posed by Clark Kimberling was "Does each generation contain a Fibonacci number or its negative?" It has been proved that every integer occurs in some G(i). - Karyn McLellan, Aug 16 2018

REFERENCES

D. Cox and K. McLellan, A problem on generation sets containing Fibonacci numbers, Fib. Quart., 55 (No. 2, 2017), 105-113.

C. Kimberling, Problem proposals, Proceedings of the Sixteenth International Conference on Fibonacci Numbers and Their Applications, P. Anderson, C. Ballot and W. Webb, eds. The Fibonacci Quarterly, 52.5(2014), 5-14.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = A005811(n) + A000523(n) for n >= 1. - Karyn McLellan, Aug 16 2018

EXAMPLE

For k = -5, n = 10 and f(10) = 7, so -5 first appears in G(7). - Karyn McLellan, Aug 16 20 18

MAPLE

f:=proc(n) option remember;

   if n <= 1 then n

   elif (n mod 2) = 0 then f(n/2)+1+((n/2) mod 2);

   else f((n-1)/2) + 2 - ((n-1)/2 mod 2); end if;

end proc;

[seq(f(n), n=0..120)];

PROG

(Python)

def A286298(n):

    if n <= 1:

        return n

    a, b = divmod(n, 2)

    return A286298(a) + 1 + b + (-1)**b*(a % 2) # Chai Wah Wu, Jun 02 2017

(PARI) a(n) = if (n==0, 0, logint(n, 2) + hammingweight(bitxor(n, n>>1))); \\ Michel Marcus, Aug 21 2018

CROSSREFS

Cf. A000523, A005811, A286299 (first differences).

Sequence in context: A021759 A070221 A020814 * A128440 A063201 A173258

Adjacent sequences:  A286295 A286296 A286297 * A286299 A286300 A286301

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jun 02 2017

STATUS

approved

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Last modified March 30 08:56 EDT 2020. Contains 333125 sequences. (Running on oeis4.)