Math::Aronson -- generate values of Aronson's sequence


 use Math::Aronson;
 my $aronson = Math::Aronson->new;
 print $aronson->next,"\n";  # 1
 print $aronson->next,"\n";  # 4
 print $aronson->next,"\n";  # 11


This is a bit of fun generating Aronson's sequence of numbers formed by self-referential occurrences of the letter T in numbers written out in words.

    T is the first, fourth, eleventh, sixteenth, ...
    ^    ^       ^      ^         ^      ^   ^
    1    4      11     16        24     29  33  <-- sequence

In the initial string "T is the", the letter T is the first and fourth letters, so those words are appended to make "T is the first, fourth". Those words have further Ts at 11 and 16, so those numbers are appended, and so on.

Spaces and punctuation are ignored. Accents like acutes are stripped for letter matching. The without_conjunctions option can ignore "and" or "et" too.


It's possible for the English sequence to end since there's no T in some numbers, but there doesn't seem enough of those, or the sequence doesn't fall on enough of them. (Is that proven?)

But for example using letter "F" instead gives a finite sequence,

    $it = Math::Aronson->new (letter => 'F');  # 1, 7 only

This is "F is the first, seventh" giving 1, 7 but ends there as there's no more "F"s in "seventh". See examples/ to run thorough which letters seem to terminate or not.


Sloane's On-Line Encyclopedia of Integer Sequences has entries for Aronson's sequence and some variations

    A005224    without_conjunctions=>1
    A055508    letter=>'H', without_conjunctions=>1
    A049525    letter=>'I', without_conjunctions=>1
    A081023    lying=>1,    without_conjunctions=>1
    A072886    lying=>1, initial_string=>"S ain't the"

    A080520    lang=>'fr'

    A081024    complement of lying A081023
    A072887    complement of lying "S ain't" A072886
    A072421    Latin P
    A072422    Latin N
    A072423    Latin T

The English sequences are without conjunctions, hence for example

    # sequence A005224
    $it = Math::Aronson->new (without_conjunctions => 1);

The "lying" versions A081023 and A072886 are presumably the same, but the sample values don't go far enough to see a difference.


The sequence is an infinite recurrence (or may be) so is generated in iterator style from an object created with various options.


$it = Math::Aronson->new (key => value, ...)

Create and return a new Aronson sequence object. The following optional key/value parameters affect the sequence.

lang => $string (default "en")

The language to use for the sequence. "en" and "fr" have defaults for the options below. Other languages can be used if you have the Lingua::Any::Numbers module.

initial_string => $str

The initial string for the sequence. The default is

    English    "T is the"
    French     "E est la"

For other languages there's no default yet and an initial_string must be given.

letter => $str

The letter to look for in the words. The default is the first letter of initial_string.

When a letter is given the default initial_string follows that, so "X is the" or "X est la".

   $it = Math::Aronson->new (letter => 'H');
   # is 1, 5, 16, 25, ...
   # per "H is the first, fifth, ..."

letter and initial_string can be given together to use a letter not at the start of the initial_string. For example,

   $it = Math::Aronson->new (letter => 'T',
                             initial_string => "I think T is");
   # is 2, 7, 21, 23, ...
   # per "I think T is second, seventh, twenty-first, ..."
without_conjunctions => $boolean (default false)

Strip conjunctions, meaning "and"s, in the wording so for instance "one hundred and four" becomes "one hundred four". The default is leave unchanged whatever conjunctions Lingua::Any::Numbers (or ordinal_func below) gives.

conjunctions_word => $string (default "and" or "et")

The conjunction word to exclude if without_conjunctions is true. The default is "and" for English or "et" for French. For other languages there's no default.

ordinal_func => $coderef (default Lingua modules)

A function to call to turn a number into words. Each call is

    $str = &$ordinal_func ($n);

The default is a call to_ordinal($n,$lang) of Lingua::Any::Numbers, or for English and French a direct call to Lingua::EN::Numbers or Lingua::FR::Numbers. The string returned can be wide chars.

An explicit ordinal_func can be used if Lingua::Any::Numbers doesn't support a desired language, or perhaps for a bit of rewording.

    $it = Math::Aronson->new
             (ordinal_func => sub {
                my ($n) = @_;
                return something_made_from($n);

There's nothing to select a gender from Lingua::Any::Numbers, as of version 0.30, so an ordinal_func might be used for instance to get feminine forms from Lingua::ES::Numbers.

lying => $bool (default false)

A "lying" version of the sequence, where the positions described and returned are those without the target letter. So for example

    T is   the   second,         third, fifth, ...
      ^^    ^^   ^^^^^^           ^
      2,3,  5,6  7,8,9,10,11,12, 14, ...      <-- sequence

Starting from "T is the", the first position is a T so "first" is not appended, but the second position is not a T so lie by giving "second", and similarly the third position, but the fourth is a T so it doesn't appear.


$n = $it->next

Return the next number in the sequence, being the next position of T (or whatever letter) in the text. The first position is 1.

If the end of the sequence has been reached then the return is an empty list (which means undef in scalar context). Because positions begin at 1 a loop can be simply

    while (my $n = $it->next) {


Accents are stripped using Unicode::Normalize if available (Perl 5.8.0 and up), or a built-in Latin-1 table as a fallback otherwise. The Latin-1 suits Lingua::FR::Numbers and probably most of the European numbers modules.

The Lingua modules and string processing means next probably isn't particularly fast. It'd be possible to go numbers-only with the usual rules for ordinals as words but generating just the positions of the "T"s or whatever desired letter, but that doesn't seem worth the effort.


Lingua::EN::Numbers, Lingua::FR::Numbers, Lingua::Any::Numbers



Math-Aronson is Copyright 2010, 2011, 2012, 2017, 2019 Kevin Ryde

Math-Aronson is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-Aronson is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-Aronson. If not, see <>.