Введение
Последовательность целых чисел
In mathematics, the look and say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221,
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11. 11 is read off as "two 1s" or 21. 21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look and say sequence was analyzed by John Conway
after he was introduced to it by one of his students at a party. The idea of the look and say sequence is similar to that of run length encoding. If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows:
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d,
Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (for d = 2, see )
В математике последовательность «смотри и говори» — это последовательность целых чисел, начинающаяся следующим образом:
In mathematics, the look and say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221,
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11. 11 is read off as "two 1s" or 21. 21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look and say sequence was analyzed by John Conway
after he was introduced to it by one of his students at a party. The idea of the look and say sequence is similar to that of run length encoding. If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows:
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d,
Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (for d = 2, see )
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, …
In mathematics, the look and say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221,
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11. 11 is read off as "two 1s" or 21. 21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look and say sequence was analyzed by John Conway
after he was introduced to it by one of his students at a party. The idea of the look and say sequence is similar to that of run length encoding. If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows:
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d,
Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (for d = 2, see )
Чтобы сгенерировать член последовательности из предыдущего члена, произнесите цифры предыдущего члена, подсчитывая количество цифр в группах одинаковых цифр. Например:
In mathematics, the look and say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221,
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11. 11 is read off as "two 1s" or 21. 21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look and say sequence was analyzed by John Conway
after he was introduced to it by one of his students at a party. The idea of the look and say sequence is similar to that of run length encoding. If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows:
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d,
Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (for d = 2, see )
1 произносится как «одна 1» или 11. 11 произносится как «две 1» или 21. 21 произносится как «одна 2, одна 1» или 1211. 1211 произносится как «одна 1, одна 2, две 1» или 111221. 111221 произносится как «три 1, две 2, одна 1» или 312211. Последовательность «смотри и говори» была проанализирована Джоном Конвеем после того, как с ней познакомил один из его студентов на вечеринке. Идея последовательности «смотри и говори» аналогична кодированию длины серии. Если начать с любой цифры d от 0 до 9, то d останется бесконечно последней цифрой последовательности. Для любого d, отличного от 1, последовательность начинается следующим образом:
In mathematics, the look and say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221,
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11. 11 is read off as "two 1s" or 21. 21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look and say sequence was analyzed by John Conway
after he was introduced to it by one of his students at a party. The idea of the look and say sequence is similar to that of run length encoding. If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows:
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d,
Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (for d = 2, see )
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, …
In mathematics, the look and say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221,
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11. 11 is read off as "two 1s" or 21. 21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look and say sequence was analyzed by John Conway
after he was introduced to it by one of his students at a party. The idea of the look and say sequence is similar to that of run length encoding. If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows:
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d,
Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (for d = 2, see )
Илан Варди назвал эту последовательность, начинающуюся с d = 3, последовательностью Конвея (для d = 2, см. ).
In mathematics, the look and say sequence is the sequence of integers beginning as follows:
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221,
To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the number of digits in groups of the same digit. For example:
1 is read off as "one 1" or 11. 11 is read off as "two 1s" or 21. 21 is read off as "one 2, one 1" or 1211. 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211. The look and say sequence was analyzed by John Conway
after he was introduced to it by one of his students at a party. The idea of the look and say sequence is similar to that of run length encoding. If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows:
d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d,
Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (for d = 2, see )
Рост
Последовательность растет неограниченно. Фактически, любой вариант, определенный началом с другого целого числа в качестве начального значения, также будет (в конечном итоге) неограниченно возрастать, за исключением вырожденной последовательности: 22, 22, 22, 22, которая остается неизменной.
Популяризация
Последовательность «смотри и говори» также широко известна как последовательность чисел Морриса, названная в честь криптографа Роберта Морриса, а головоломка «Какое число следующее в последовательности 1, 11, 21, 1211, 111221?» иногда называется «Яйцом кукушки», по мотивам описания Морриса в книге Клиффорда Столла «Яйцо кукушки».
Вариации
Существует множество возможных вариаций правила, используемого для генерации последовательности "смотри и говори". Например, для формирования "модели гороха" необходимо прочитать предыдущий член последовательности и посчитать все вхождения каждой цифры, перечисляя их в порядке первого появления, а не только те, которые встречаются в непрерывном блоке. Начиная с начального значения 1, последовательность "модели гороха" выглядит следующим образом: 1, 11 ("один 1"), 21 ("два 1"), 1211 ("один 2 и один 1"), 3112 ("три 1 и один 2"), 132112 ("один 3, два 1 и один 2"), 311322 ("три 1, один 3 и два 2") и так далее. Эта версия "модели гороха" в конечном итоге образует цикл с двумя "атомными" членами: 23322114 и 32232114. Возможны и другие варианты "модели гороха"; например, вместо чтения цифр в порядке их первого появления, можно читать их в порядке возрастания. В этом случае, член последовательности, следующий за 21, будет 1112 ("один 1, один 2"), а член, следующий за 3112, будет 211213 ("два 1, один 2 и один 3"). Эти последовательности существенно отличаются от последовательности "смотри и говори". Важно отметить, что в отличие от последовательностей Конвея, данный член "модели гороха" не однозначно определяет предыдущий член. Более того, для любого начального значения "модель гороха" генерирует члены ограниченной длины: эта граница обычно не превышает 2 × Radix + 2 цифр (22 цифры для десятичной системы счисления) и может превышать 3 × Radix (30 цифр для десятичной системы счисления) только для длинных, вырожденных начальных значений (последовательность из "100 единиц" и т.п.). В этих крайних случаях отдельные элементы десятичных последовательностей немедленно переходят в перестановку вида, где буквы a–j обозначают счетчики цифр из предыдущего элемента последовательности. Поскольку последовательность бесконечна, а длина каждого элемента ограничена, она неизбежно должна повториться в силу принципа Дирихле. Следовательно, последовательности "модели гороха" всегда в конечном итоге становятся периодическими.