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#!/bin/bash
# Douglas Hofstadter's notorious "Q-series":
# Q(1) = Q(2) = 1
# Q(n) = Q(n - Q(n-1)) + Q(n - Q(n-2)), for n>2
# This is a "chaotic" integer series with strange
#+ and unpredictable behavior.
# The first 20 terms of the series are:
# 1 1 2 3 3 4 5 5 6 6 6 8 8 8 10 9 10 11 11 12
# See Hofstadter's book, _Goedel, Escher, Bach: An Eternal Golden Braid_,
#+ p. 137, ff.
LIMIT=100 # Number of terms to calculate.
LINEWIDTH=20 # Number of terms printed per line.
Q[1]=1 # First two terms of series are 1.
Q[2]=1
echo
echo "Q-series [$LIMIT terms]:"
echo -n "${Q[1]} " # Output first two terms.
echo -n "${Q[2]} "
for ((n=3; n <= $LIMIT; n++)) # C-like loop expression.
do # Q[n] = Q[n - Q[n-1]] + Q[n - Q[n-2]] for n>2
# Need to break the expression into intermediate terms,
#+ since Bash doesn't handle complex array arithmetic very well.
let "n1 = $n - 1" # n-1
let "n2 = $n - 2" # n-2
t0=`expr $n - ${Q[n1]}` # n - Q[n-1]
t1=`expr $n - ${Q[n2]}` # n - Q[n-2]
T0=${Q[t0]} # Q[n - Q[n-1]]
T1=${Q[t1]} # Q[n - Q[n-2]]
Q[n]=`expr $T0 + $T1` # Q[n - Q[n-1]] + Q[n - Q[n-2]]
echo -n "${Q[n]} "
if [ `expr $n % $LINEWIDTH` -eq 0 ] # Format output.
then # ^ modulo
echo # Break lines into neat chunks.
fi
done
echo
exit 0
# This is an iterative implementation of the Q-series.
# The more intuitive recursive implementation is left as an exercise.
# Warning: calculating this series recursively takes a VERY long time
#+ via a script. C/C++ would be orders of magnitude faster.
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