A318058 - OEIS

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A318058

a(n) is the number of decimal places to which the n-th convergent of the continued fraction expansion of the golden section matches the correct value.

0, -1, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 4, 5, 5, 5, 6, 5, 7, 7, 8, 7, 9, 9, 9, 10, 10, 10, 11, 10, 12, 12, 13, 12, 13, 14, 15, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 18, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 23, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 29, 30, 30

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,7

COMMENTS

The correct decimal value of the golden section is given in A001622; the continued fraction terms of the golden section is given in A000012.

For the number of correct decimal digits of the golden section see A318057.

The denominator of the k-th convergent obtained from a continued fraction tend to k*A001622; the error between the k-th convergent and the constant itself tends to 1/(2*k*A001622), or in binary digits 2*k*log(A001622)/log(2) bits after the binary point.

LINKS

Table of n, a(n) for n=1..75.

FORMULA

Limit_{n -> oo} a(n)/n = 2*log(A001622)/log(10) = 2*A002390/log(10) = A202543/log(10) = 2*A097348.

EXAMPLE

n convergent decimal expansion a(n)

== ============= ========================= ====

1 1 / 1 1.0 0

2 2 / 1 2.0 -1

3 3 / 2 1.5 0

4 5 / 3 1.66 1

5 8 / 5 1.60 1

6 13 / 8 1.62 1

7 21 / 13 1.615 2

8 34 / 21 1.619 2

9 55 / 34 1.617 2

10 89 / 55 1.6181 3

oo lim = A001622 1.6180339887498948482 --

PROG

(Python)

p, q, i, base = 1, 1, 0, 10

while i < 20200:

....p, q, i = p+q, p, i+1

a0, p, q = p//q, q, p

i, p, dd = 0, p*base, [0]

while i < 30000:

....d, p, i = p//q, (p%q)*base, i+1

....dd = dd+[d]

n, pn, qn = 0, 1, 0

while n < 20000:

....n, pn, qn = n+1, pn+qn, pn

....if pn//qn != a0:

........print(n, "- manual!")

....else:

........i, p, q, di = 0, (pn%qn)*base, qn, 0

........while di == dd[i]:

............i, di, p = i+1, p//q, (p%q)*base

........print(n, i-1)

CROSSREFS

Cf. A000012, A001622, A002390, A097348, A202543, A318057.

Sequence in context: A332888 A258596 A273784 * A345901 A305434 A319478

Adjacent sequences: A318055 A318056 A318057 * A318059 A318060 A318061

KEYWORD

sign,base

AUTHOR

A.H.M. Smeets, Aug 14 2018

STATUS

approved