A045918 Describe n. Also called the "Say What You See" or "Look and Say" sequence LS(n).

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1110, 21, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1210, 1211, 22, 1213, 1214, 1215, 1216, 1217, 1218, 1219, 1310, 1311, 1312, 23, 1314, 1315, 1316, 1317, 1318, 1319, 1410, 1411, 1412, 1413, 24, 1415, 1416, 1417

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• base

Comments

a(1111111111) = a((10^10 - 1)/9) = 101 is the first term with an odd number of digits; 3-digit terms are unambiguous, but already the 2nd 4-digit term is LS( 12 ) = 1112 = LS( 2*(10^111-1)/9 ) ("hundred eleven 2's"). The smallest n such that LS(n) = LS(k) for some k < n (i.e. the largest n such that the restriction of LS to [0..n-1] is injective) appears to be 10*(10^11 - 1)/9 : LS(eleven '1's, one '0') = 11110 = LS(one '1', eleven '0's). - M. F. Hasler, Nov 14 2006

A121993 gives numbers m such that a(m) < m. - Reinhard Zumkeller, Jan 25 2014

Examples

			23 has "one 2, one 3", so a(23) = 1213.
		

References

• J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Computation, Springer, NY 1987, pp. 173-188.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Kevin Watkins, Abstract Interpretation Using Laziness: Proving Conway's Lost Cosmological Theorem,
• Kevin Watkins, Proving Conway's Lost Cosmological Theorem, POP seminar talk, CMU, Dec 2006
• Eric Weisstein's World of Mathematics, Look and Say Sequence
• Wikipedia, Look-and-say sequence

Crossrefs

Cf. A005150. See also A056815.

Programs

Haskell

-- see Watkins link, p. 3.
import Data.List (unfoldr, group); import Data.Tuple (swap)
a045918 0 = 10
a045918 n = foldl (\v d -> 10 * v + d) 0 $ say $ reverse $ unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 10) n
   where say = concat . map code . group
         code xs = [toInteger $ length xs, head xs]
-- Reinhard Zumkeller, Aug 09 2012

Maple

LS:=n-> if n>9 then LS(op(convert(n,base,10))) else for i from 2 to nargs do if args[i] <> n then RETURN(( LS( args[i..nargs] )*10^length(i-1) + i-1)*10 + n ) fi od: 10*nargs + n fi; # M. F. Hasler, Nov 14 2006

Mathematica

LookAndSayA[n_]  := FromDigits@ Flatten@ IntegerDigits@ Flatten[ Through[{Length, First}[#]] & /@ Split@ IntegerDigits@ n] (* Robert G. Wilson v, Jan 27 2012 *)

PARI

A045918(a)={my(c=1);for(j=2,#a=Vec(Str(a)),if(a[j-1]==a[j],a[j-1]="";c++,a[j-1]=Str(c,a[j-1]);c=1));a[#a]=Str(c,a[#a]);eval(concat(a))}  \\ M. F. Hasler, Jan 27 2012

Python

from re import finditer
def A045918(n):
    return int(''.join([str(len(m.group(0)))+m.group(0)[0] for m in finditer(r'(\d)\1*',str(n))]))
# Chai Wah Wu, Dec 03 2014

Python

from itertools import groupby
def LS(n): return int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
print([LS(n) for n in range(48)]) # Michael S. Branicky, Jul 27 2022

Extensions

Added Mma program from A056815. - N. J. A. Sloane, Feb 02 2012