A | B | C | D | E | F | G | H | CH | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
In mathematics, the look-and-say sequence is the sequence of integers beginning as follows:
- 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ... (sequence A005150 in the OEIS).
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[1] after he was introduced to it by one of his students at a party.[2][3]
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 (sequence A006715 in the OEIS). (for d = 2, see OEIS: A006751)[4]
Basic properties
Growth
The sequence grows indefinitely. In fact, any variant defined by starting with a different integer seed number will (eventually) also grow indefinitely, except for the degenerate sequence: 22, 22, 22, 22, ... which remains the same size.(sequence A010861 in the OEIS)[5]
Digits presence limitation
No digits other than 1, 2, and 3 appear in the sequence, unless the seed number contains such a digit or a run of more than three of the same digit.[5]
Cosmological decay
Conway's cosmological theorem asserts that every sequence eventually splits ("decays") into a sequence of "atomic elements", which are finite subsequences that never again interact with their neighbors. There are 92 elements containing the digits 1, 2, and 3 only, which John Conway named after the 92 naturally-occurring chemical elements up to uranium, calling the sequence audioactive. There are also two "transuranic" elements (Np and Pu) for each digit other than 1, 2, and 3.[5][6] Below is a table of all such elements:
| All "atomic elements" (Where Ek is included within the derivate of Ek+1 except Np and Pu)[1] | ||||
|---|---|---|---|---|
| Atomic number (n) | Element name (Ek) | Sequence | Decays into[5] | Abundance |
| 1 | H | 22 | H | 91790.383216 |
| 2 | He | 13112221133211322112211213322112 | Hf.Pa.H.Ca.Li | 3237.2968588 |
| 3 | Li | 312211322212221121123222112 | He | 4220.0665982 |
| 4 | Be | 111312211312113221133211322112211213322112 | Ge.Ca.Li | 2263.8860325 |
| 5 | B | 1321132122211322212221121123222112 | Be | 2951.1503716 |
| 6 | C | 3113112211322112211213322112 | B | 3847.0525419 |
| 7 | N | 111312212221121123222112 | C | 5014.9302464 |
| 8 | O | 132112211213322112 | N | 6537.3490750 |
| 9 | F | 31121123222112 | O | 8521.9396539 |
| 10 | Ne | 111213322112 | F | 11109.006696 |
| 11 | Na | 123222112 | Ne | 14481.448773 |
| 12 | Mg | 3113322112 | Pm.Na | 18850.441228 |
| 13 | Al | 1113222112 | Mg | 24573.006696 |
| 14 | Si | 1322112 | Al | 32032.812960 |
| 15 | P | 311311222112 | Ho.Si | 14895.886658 |
| 16 | S | 1113122112 | P | 19417.939250 |
| 17 | Cl | 132112 | S | 25312.784218 |
| 18 | Ar | 3112 | Cl | 32997.170122 |
| 19 | K | 1112 | Ar | 43014.360913 |
| 20 | Ca | 12 | K | 56072.543129 |
| 21 | Sc | 3113112221133112 | Ho.Pa.H.Ca.Co | 9302.0974443 |
| 22 | Ti | 11131221131112 | Sc | 12126.002783 |
| 23 | V | 13211312 | Ti | 15807.181592 |
| 24 | Cr | 31132 | V | 20605.882611 |
| 25 | Mn | 111311222112 | Cr.Si | 26861.360180 |
| 26 | Fe | 13122112 | Mn | 35015.858546 |
| 27 | Co | 32112 | Fe | 45645.877256 |
| 28 | Ni | 11133112 | Zn.Co | 13871.123200 |
| 29 | Cu | 131112 | Ni | 18082.082203 |
| 30 | Zn | 312 | Cu | 23571.391336 |
| 31 | Ga | 13221133122211332 | Eu.Ca.Ac.H.Ca.Zn | 1447.8905642 |
| 32 | Ge | 31131122211311122113222 | Ho.Ga | 1887.4372276 |
| 33 | As | 11131221131211322113322112 | Ge.Na | 27.246216076 |
| 34 | Se | 13211321222113222112 | As | 35.517547944 |
| 35 | Br | 3113112211322112 | Se | 46.299868152 |
| 36 | Kr | 11131221222112 | Br | 60.355455682 |
| 37 | Rb | 1321122112 | Kr | 78.678000089 |
| 38 | Sr | 3112112 | Rb | 102.56285249 |
| 39 | Y | 1112133 | Sr.U | 133.69860315 |
| 40 | Zr | 12322211331222113112211 | Y.H.Ca.Tc | 174.28645997 |
| 41 | Nb | 1113122113322113111221131221 | Er.Zr | 227.19586752 |
| 42 | Mo | 13211322211312113211 | Nb | 296.16736852 |
| 43 | Tc | 311322113212221 | Mo | 386.07704943 |
| 44 | Ru | 132211331222113112211 | Eu.Ca.Tc | 328.99480576 |
| 45 | Rh | 311311222113111221131221 | Ho.Ru | 428.87015041 |
| 46 | Pd | 111312211312113211 | Rh | 559.06537946 |
| 47 | Ag | 132113212221 | Pd | 728.78492056 |
| 48 | Cd | 3113112211 | Ag | 950.02745646 |
| 49 | In | 11131221 | Cd | 1238.4341972 |
| 50 | Sn | 13211 | In | 1614.3946687 |
| 51 | Sb | 3112221 | Pm.Sn | 2104.4881933 |
| 52 | Te | 1322113312211 | Eu.Ca.Sb | 2743.3629718 |
| 53 | I | 311311222113111221 | Ho.Te | 3576.1856107 |
| 54 | Xe | 11131221131211 | I | 4661.8342720 |
| 55 | Cs | 13211321 | Xe | 6077.0611889 |
| 56 | Ba | 311311 | Cs | 7921.9188284 |
| 57 | La | 11131 | Ba | 10326.833312 |
| 58 | Ce | 1321133112 | La.H.Ca.Co | 13461.825166 |
| 59 | Pr | 31131112 | Ce | 17548.529287 |
| 60 | Nd | 111312 | Pr | 22875.863883 |
| 61 | Pm | 132 | Nd | Zdroj:https://en.wikipedia.org?pojem=Look-and-say_sequence|