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

%I A045918 #46 Feb 16 2025 08:32:38
%S A045918 10,11,12,13,14,15,16,17,18,19,1110,21,1112,1113,1114,1115,1116,1117,
%T A045918 1118,1119,1210,1211,22,1213,1214,1215,1216,1217,1218,1219,1310,1311,
%U A045918 1312,23,1314,1315,1316,1317,1318,1319,1410,1411,1412,1413,24,1415,1416,1417
%N A045918 Describe n. Also called the "Say What You See" or "Look and Say" sequence LS(n).
%C A045918 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
%C A045918 A121993 gives numbers m such that a(m) < m. - _Reinhard Zumkeller_, Jan 25 2014
%D A045918 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.
%H A045918 Reinhard Zumkeller, <a href="/A045918/b045918.txt">Table of n, a(n) for n = 0..10000</a>
%H A045918 Kevin Watkins, <a href="http://www.cs.cmu.edu/~kw/pubs/conway.pdf">Abstract Interpretation Using Laziness: Proving Conway's Lost Cosmological Theorem, </a>
%H A045918 Kevin Watkins, <a href="http://www.cs.cmu.edu/~kw/pubs/conwayslides.pdf">Proving Conway's Lost Cosmological Theorem, POP seminar talk, CMU, Dec 2006</a>
%H A045918 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LookandSaySequence.html">Look and Say Sequence</a>
%H A045918 Wikipedia, <a href="http://en.wikipedia.org/wiki/Look-and-say_sequence">Look-and-say sequence</a>
%e A045918 23 has "one 2, one 3", so a(23) = 1213.
%p A045918 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
%t A045918 LookAndSayA[n_]  := FromDigits@ Flatten@ IntegerDigits@ Flatten[ Through[{Length, First}[#]] & /@ Split@ IntegerDigits@ n] (* _Robert G. Wilson v_, Jan 27 2012 *)
%o A045918 (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
%o A045918 (Haskell) -- see Watkins link, p. 3.
%o A045918 import Data.List (unfoldr, group); import Data.Tuple (swap)
%o A045918 a045918 0 = 10
%o A045918 a045918 n = foldl (\v d -> 10 * v + d) 0 $ say $ reverse $ unfoldr
%o A045918    (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 10) n
%o A045918    where say = concat . map code . group
%o A045918          code xs = [toInteger $ length xs, head xs]
%o A045918 -- _Reinhard Zumkeller_, Aug 09 2012
%o A045918 (Python)
%o A045918 from re import finditer
%o A045918 def A045918(n):
%o A045918     return int(''.join([str(len(m.group(0)))+m.group(0)[0] for m in finditer(r'(\d)\1*',str(n))]))
%o A045918 # _Chai Wah Wu_, Dec 03 2014
%o A045918 (Python)
%o A045918 from itertools import groupby
%o A045918 def LS(n): return int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
%o A045918 print([LS(n) for n in range(48)]) # _Michael S. Branicky_, Jul 27 2022
%Y A045918 Cf. A005150. See also A056815.
%K A045918 nonn,base
%O A045918 0,1
%A A045918 _N. J. A. Sloane_
%E A045918 Added Mma program from A056815. - _N. J. A. Sloane_, Feb 02 2012