A317907 Number of binary places to which n-th convergent of continued fraction expansion of Khintchine's constant matches the correct value.
0, -1, 5, 3, 9, 8, 12, 14, 16, 22, 25, 27, 30, 33, 39, 44, 42, 49, 52, 51, 56, 55, 64, 70, 73, 77, 81, 83, 82, 85, 88, 92, 93, 99, 101, 104, 109, 104, 111, 114, 117, 120, 122, 124, 126, 129, 131, 133, 136, 139, 138, 144, 138, 148, 151, 150, 153, 156, 158, 162
Offset: 1
Examples
n convergent binary expansion a(n) == ============= ========================== ==== 1 2 / 1 10.0... 0 2 3 / 1 11.0... -1 3 8 / 3 10.101010... 5 4 43 / 16 10.1011... 3 5 51 / 19 10.1010111100... 9 oo lim = A317906 10.101011110111100111... --
Links
- A.H.M. Smeets, Table of n, a(n) for n = 1..19999
Programs
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Python
i,cf = 0,[] while i <= 20100: c = A002211(i) cf,i = cf+[c],i+1 p0,p1,q0,q1,i,base = cf[0],1,1,0,1,2 while i <= 20100: p0,p1,q0,q1,i = cf[i]*p0+p1,p0,cf[i]*q0+q1,q0,i+1 a0 = p0//q0 p0 = p0-a0*q0 i,p0,dd = 0,p0*base,[a0] while i < 70000: d,p0,i = p0//q0,(p0%q0)*base,i+1 dd = dd+[d] n,pn0,pn1,qn0,qn1 = 1,a0,1,1,0 while n <= 20000: p,q = pn0,qn0 if p//q != a0: print(n,"- manual!") else: i,p,di = 0,(p%q)*base,a0 while di == dd[i]: i,di,p = i+1,p//q,(p%q)*base print(n,i-1) n,pn0,pn1,qn0,qn1 = n+1,cf[n]*pn0+pn1,pn0,cf[n]*qn0+qn1,qn0
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