|
| |
|
|
A005150
|
|
Look and Say sequence: describe the previous term! (method A - initial term is 1).
(Formerly M4780)
|
|
117
|
|
|
|
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, 11131221133112132113212221, 3113112221232112111312211312113211
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
Method A = 'frequency' followed by 'digit'-indication.
Also known as the "Say What You see" sequence.
Only the digits 1, 2 and 3 appear in any term. - Robert G. Wilson v Jan 22 2004.
All terms end by 1 (the seed) and, except the third a(3), begin by 1 or 3. [Jean-Christophe Hervé, May 07 2013]
Proof that 333 never appears in any a(n): suppose it appears for the first time in a(n); because of 'three 3' in 333, it would imply that 333 is also in a(n-1), which is a contradiction. [Jean-Christophe Hervé, May 09 2013]
|
|
|
REFERENCES
|
J. H. Conway, The weird and wonderful chemistry of audioactive decay, Eureka 46 (1986) 5-16.
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.
S. B. Ekhad and D. Zeilberger, Proof of Conway's lost cosmological theorem, Elect. Res. Announcements Amer, Math. Soc., 3 (1997), 78-82.
S. Eliahou and M. J. Erickson, Mutually describing multisets and integer partitions, Discrete Mathematics, Volume 313, Issue 4, 28 February 2013, Pages 422-433. - From N. J. A. Sloane, Jan 03 2013
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452-455.
M. Gilpin, On the generalized Gleichniszahlen-Reihe sequence, Manuscript, Jul 05 1994.
X. Gourdon and B. Salvy, http://pauillac.inria.fr/algo/papers/html/GoSa96/GoSa96.html Effective asymptotics of linear recurrences with rational coefficients, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163. See p. 161.
M. Hilgemeier, Der Gleichniszahlen-Reihe, Bild der Wissenschaft, 12 (1986), 194-195.
A. Lakhtakia and C. Pickover, Observations on the Gleichniszahlen-Reihe: An Unusual Number Theory Sequence, J. Recreational Math., 25 (No. 3, 1993), 192-198.
O. Martin, Look-and-Say Biochemistry: Exponential RNA and Multistranded DNA, Amer. Math. Monthly, 113 (No. 4, 2006), 289-307.
M. Oscar, Look-and-say biochemistry ..., Amer. Math. Monthly, 113 (2006), 289-307. - From N. J. A. Sloane, Feb 19 2013
Clifford A. Pickover, Computers and the Imagination, St Martin's Press, NY, 1991.
Clifford A. Pickover, Fractal horizons : the future use of fractals, New York : St. Martin's Press, 1996. ISBN 0312125992. Chapter 7 has an extensive description of the elements and their properties.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 486.
J. Sauerberg and L. Shu, The long and the short on counting sequences, Amer. Math. Monthly, 104 (1997), 306-317.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, 1999, p. 23.
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 4.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n = 1..25
Henry Bottomley, Evolution of Conway's 92 Look and Say audioactive elements
S. R. Finch, Conway's Constant
M. Hilgemeier, One metaphor fits all, in Fractal Horizons, ed. C. A Pickover, St. Martins, NY, 1996, pp. 137-161.
R. A. Litherland, The audioactive package
R. A. Litherland, Conway's cosmological theorem
M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 37, etc.
Paulo Ortolan, Java program for A005150
T. Sillke, Conway sequence
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
D. Zeilberger, [math/9808077] Proof of Conway's Lost Cosmological Theorem
D. Zeilberger, Proof of Conway's lost cosmological theorem, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 78-82.
|
|
|
FORMULA
|
a(n+1) = A045918(a(n)). - Reinhard Zumkeller, Aug 09 2012
a(n) = sum{A034002(n,k)*10^(A005341(n)-k): k=1..A005341(n)}. - Reinhard Zumkeller, Dec 15 2012
|
|
|
EXAMPLE
|
E.g. the term after 1211 is obtained by saying "one 1, one 2, two 1's", which gives 111221.
|
|
|
MATHEMATICA
|
RunLengthEncode[ x_List ] := (Through[ {First, Length}[ #1 ] ] &) /@ Split[ x ]; LookAndSay[ n_, d_:1 ] := NestList[ Flatten[ Reverse /@ RunLengthEncode[ # ] ] &, {d}, n - 1 ]; F[ n_ ] := LookAndSay[ n, 1 ][ [ n ] ]; Table[ FromDigits[ F[ n ] ], {n, 1, 15} ]
A005150[1]:=1; A005150[n_] := A005150[n] = FromDigits[Flatten[{Length[#], First[#]}&/@Split[IntegerDigits[A005150[n-1]]]]]; Map[A005150, Range[25]] (* Peter J. C. Moses, Mar 21 2013 *)
|
|
|
PROG
|
(Haskell)
import List
say :: Integer -> Integer
say = read . concatMap saygroup . group . show
where saygroup s = (show $ length s) ++ [head s]
look_and_say :: [Integer]
look_and_say = 1 : map say look_and_say
-- Josh Triplett (josh(AT)freedesktop.org), Jan 03 2007
(Haskell)
a005150 = foldl1 (\v d -> 10 * v + d) . map toInteger . a034002_row
-- Reinhard Zumkeller, Aug 09 2012
(Java) See Paulo Ortolan link.
(PERL)
#!/usr/bin/perl
$str="1"; for (1 .. shift(@ARGV)) { print($str, ", "); @a = split(//, $str); $str=""; $nd=shift(@a); while (defined($nd)) { $d=$nd; $cnt=0; while (defined($nd) && ($nd eq $d)) { $cnt++; $nd = shift(@a); } $str .= $cnt.$d; } } print($str); # Jeff Quilici (jeff(AT)quilici.com), Aug 12 2003
(PERL) ## better program from Arne 'Timwi' Heizmann (timwi(AT)gmx.net), Mar 12 2008)
# This outputs the first n elements of the sequence, where n is given on the command line.
$s = 1;
for (2..shift @ARGV) {
print "$s, ";
$s =~ s/(.)\1*/(length $&).$1/eg;
}
print "$s\n";
(Python, from Olivier Mengue (dolmen(AT)users.sourceforge.net), Jul 01 2005: replace leading dots by blanks before running)
.def A005150(n):
... p = "1"
... seq = [1]
... while (n > 1):
....... q = ''
....... idx = 0 # Index
....... l = len(p) # Length
....... while idx < l:
........... start = idx
........... idx = idx + 1
........... while idx < l and p[idx] == p[start]:
............... idx = idx + 1
........... q = q + str(idx-start) + p[start]
....... n, p = n - 1, q
....... seq.append(int(p))
... return seq
(A second Python program from E. Johnson (ejohnso9(AT)earthlink.net), Mar 31 2008: replace leading dots by blanks before running)
.def A005150(n):
... seq = [1] + [None] * (n - 1) # allocate entire array space
... def say(s):
....... acc = '' # initialize accumulator
....... while len(s) > 0:
........... i = 0
........... c = s[0] # char of first run
........... while (i < len(s) and s[i] == c): # scan first digit run
............... i += 1
........... acc += str(i) + c # append description of first run
........... if i == len(s):
............... break # done
........... else:
............... s = s[i:] # trim leading run of digits
....... return acc
... for i in xrange(1, n):
....... seq[i] = int(say(str(seq[i-1])))
... return seq
(PARI) A005150(n, a=1)={ while(n--, 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]); a=concat(a)); a } \\ - M. F. Hasler, Jun 30 2011
|
|
|
CROSSREFS
|
Cf. A001155, A006751, A006715, A001140, A001141, A001143, A001145, A001151, A001154, A007651.
Cf. A001387. Length of n-th term = A005341. Periodic table: A119566.
Cf. A225224, A221646, A225212 (continuous versions).
Sequence in context: A158081 A007890 A063850 * A001388 A110393 A202627
Adjacent sequences: A005147 A005148 A005149 * A005151 A005152 A005153
|
|
|
KEYWORD
|
nonn,base,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
