

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
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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), 105113.
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), 514.


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((n1)/2) + 2  ((n1)/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



