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A119719 Continued fraction expansion of the value (mod 1) where ?(x)-x attains its global maximum. 0
0, 1, 3, 1, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

?(x) is Minkowski's question mark function. Note ?(x)-x is odd and has period 1. Finding the maximum of ?(x)-x difficult; fractal local maxima abound. Given that this continued fraction expansion represents the real a, we note the global minimum of ?(x)-x occurs (symmetrically across x=1/2) at 1-a (mod 1). We expect even entries will remain near (if not at) 1 and odd entries will grow very slowly (perhaps not monotonically).

A lookahead algorithm scanning over continued fractions with coefficients <= 20 and lookahead depth 4 returns the fraction [0; 1, 3, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, ..]. This corresponds to an x value of 0.7928941486060[1], at which point ?(x)-x is equal to 0.1425907067997[2]. - Charlie Neder, Oct 27 2018

LINKS

Table of n, a(n) for n=0..4.

Index entries for Minkowski's question mark function

Index entries for sequences related to Minkowski's question mark function

EXAMPLE

a = [0;1,3,1,4,..?..]

PROG

(Python) from itertools import product

def qx(arr): #given continued fraction,

..qx = 0

..for i in range(1, len(arr)): #generate ?(x)/2

....qx += (-1)**(i+1) / 2**sum(arr[:i+1])

..ratio = arr[-1]

..for i in range(len(arr)-2, -1, -1): #generate x

....ratio = arr[i] + 1/ratio

..return 2*qx - ratio, ratio #subtract

arr = [0, 1]

for k in range(1, 19):

..cap = [0, ()] #current best branch

..for tag in product(range(1, 21), repeat=4):

....res = qx(arr + list(tag)) #test a branch, record if best

....if res[0] > cap[0]: cap = [res[0], tag, res[1]]

..print(cap) #print current ?(x)-x, best branch, current x

..arr.append(cap[1][0]) #go down the branch

# Charlie Neder, Oct 27 2018

CROSSREFS

Sequence in context: A229755 A257634 A110790 * A125162 A174382 A123730

Adjacent sequences:  A119716 A119717 A119718 * A119720 A119721 A119722

KEYWORD

cofr,hard,more,nonn

AUTHOR

Joseph Biberstine (jrbibers(AT)indiana.edu), Jun 13 2006

STATUS

approved

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Last modified September 21 11:18 EDT 2019. Contains 327253 sequences. (Running on oeis4.)