A317908 Number of decimal places to which the n-th convergent of the continued fraction expansion of Khintchine's constant matches the correct value.
0, -1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 8, 8, 9, 11, 13, 12, 14, 15, 16, 16, 16, 18, 21, 21, 23, 24, 24, 25, 25, 26, 27, 28, 29, 30, 30, 32, 32, 33, 33, 36, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 44, 46, 47, 48, 48, 49, 50, 51, 54, 55, 56, 56, 58, 58, 60
Offset: 1
Examples
n convergent decimal expansion a(n) == ============= ==================== ==== 1 2 / 1 2.0 0 2 3 / 1 3.0 -1 3 8 / 3 2.66... 1 4 43 / 16 2.687... 2 5 51 / 19 2.684... 2 6 94 / 35 2.6857... 3 7 239 / 89 2.6853... 3 8 333 / 124 2.68548... 4 9 572 / 213 2.68544... 4 10 2049 / 763 2.6854521... 6 oo lim = A002210 2.685452001065306... --
Links
- A.H.M. Smeets, Table of n, a(n) for n = 1..20000
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],1,1,0,1,10 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 < 21000: 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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