A140129 Let c(n) = 0 if n=1, (c(floor(n/2)) + c(floor((n+1)/2)))/2 if 1 < n < 3*2^floor(log_2(n)-1), and c(n-2^floor(log_2(n))) + 1 otherwise. Then a(n) = numerator(c(n)).

0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 3, 1, 3, 2, 3, 0, 1, 1, 3, 1, 5, 3, 7, 1, 5, 3, 7, 2, 5, 3, 4, 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 9, 5, 11, 3, 13, 7, 15, 2, 9, 5, 11, 3, 7, 4, 5, 0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31

Offset: 1

Author

Reinhard Zumkeller, May 14 2008

Keywords

• nonn
• frac
• look

Comments

C(k) = {a(n)/A140130(n): 2^(k-1) <= n < 2^k} = nonnegative Conway numbers created on day k according to the genesis reported by Knuth.

c(2^n-1) = n-1; c(2^n) = 0; c(3*2^n) = 1; c(5*2^n) = 1/2;

for n>1: a(A023758(n)) = A002262(n-2) and A140130(A023758(n))=1;

a(n) = a(n - 2^floor(log_2(n)) + A140130(n - 2^floor(log_2(n)) for n with 3*2^floor(log_2(n)-1) <= n < 2^floor(log_2(n)).

Examples

			C(1) = {0};
C(2) = {0, 1};
C(3) = {0, 1/2, 1,2};
C(4) = {0, 1/4, 1/2, 3/4, 1, 3/2, 2, 3};
C(5) = {0, 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, 1, 5/4, 3/2, 7/4, 2, 5/2, 3, 4}.
		

References

• D. E. Knuth, Surreal Numbers, Addison-Wesley, Reading, 1974.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..8191
• Wikipedia, Surreal number

Crossrefs

Cf. A000523, A007283.