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

0,1

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

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

Example

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

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A005150. See also A056815.

Sequence in context: A047842 A047843 A097598 * A244112 A088476 A248014

Adjacent sequences: A045915 A045916 A045917 * A045919 A045920 A045921

Keyword

nonn,base

Author

N. J. A. Sloane

Extensions

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

Status

approved