A228162 Number of bangbangs (!!) in shell substitution when starting with : '!!' and : "!!" '!!'.

1, 2, 2, 3, 5, 17, 161, 15681, 159591041, 16866847940875521, 189345699699803478502456213711361

Offset: 0

Author

Volker Diels-Grabsch, Aug 16 2013

Keywords

• nonn

Comments

Executing the following no-op commands in a unix shell (bash, dash or zsh)

: '!!'

: "!!" '!!'

followed by repeated execution of the last command (arrow-up, enter) leads to a series of substitutions due to the !! (bangbang) operator:

: ": '!!'" '!!'

: ": ': ": '!!'" '!!''" '!!'

: ": ': ": '!!'" ': ": ': ": '!!'" '!!''" '!!'''" '!!'

These commands show a very irregular pattern as some of the single-quotes (') and double quotes(") switch their role from string delimiters to string contents and vice versa.

Also, these commands grow very fast. After a few repetitions, they'll quickly fill the RAM as well as the hard disk (because of the command history file).

This sequence describes the number of bangbangs (!!) of each command.

Examples

			For n = 4 the value a(4) = 5 is the number of bangbangs (!!) in the following command:
: ": ': ": '!!'" ': ": ': ": '!!'" '!!''" '!!'''" '!!'
		

Links

• Volker Diels-Grabsch, Table of n, a(n) for n = 0..14
• Shadab Ahmed, Bang Bang !!
• Robin Houston, The algebra of Unix command substitution

Programs

Python

for i, ai in enumerate([1, 2, 2, 3, 5]):
    print('%d %d' % (i, ai))
p, q = 2, 4
for i in range(5, 15):
    print('%d %d' % (i, 4*p + 2*q + 1))
    p, q = 3*p*p + 2*p + p*q, 2*p*p + 2*p*q + 2*q

Formula

a(0) = 1, a(1) = 2, a(2) = 2, a(3) = 3, a(4) = 5, and for n >= 5, a(n) = 4*p(n) + 2*q(n) + 1 where p(5) = 2, q(5) = 4, and p(n+1) = 3*p(n)^2 + 2*p(n) + p(n)*q(n) and q(n+1) = 2*p(n)^2 + 2*p(n)*q(n) + 2*q(n).