A220424 -id:A220424 - cp's OEIS frontend

A007651 Describe the previous term! (method B - initial term is 1).

1, 11, 12, 1121, 122111, 112213, 12221131, 1123123111, 12213111213113, 11221131132111311231, 12221231123121133112213111, 1123112131122131112112321222113113, 1221311221113112221131132112213121112312311231

Offset: 1

N. J. A. Sloane

Method B = 'digit'-indication followed by 'frequency'.

			The term after 1121 is obtained by saying "1 twice, 2 once, 1 once", which gives 122111.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..25

Cf. A005150, A022470, A022499, A022500-A022505.

a007651 = foldl1 (\v d -> 10 * v + d) . map toInteger . a220424_row
-- Reinhard Zumkeller, Dec 15 2012

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[ Reverse[ F[ n ] ] ], {n, 1, 15} ]
a[1] = 1; a[n_] := a[n] = FromDigits[Flatten[{First[#], Length[#]}&/@Split[IntegerDigits[a[n-1]]]]]; Map[a, Range[25]] (* Peter J. C. Moses, Mar 22 2013 *)

from itertools import accumulate, groupby, repeat
def summarize(n, _): return int("".join(k+str(len(list(g))) for k, g in groupby(str(n))))
def aupto(terms): return list(accumulate(repeat(1, terms), summarize))
print(aupto(13)) # Michael S. Branicky, Sep 18 2022

a(n) = Sum_{k=1..A005341(n)} A220424(n,k)*10^(A005341(n)-k). - Reinhard Zumkeller, Dec 15 2012

A005341 Length of n-th term in Look and Say sequences A005150 and A007651.

Original entry on oeis.org

1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, 78, 102, 134, 176, 226, 302, 408, 528, 678, 904, 1182, 1540, 2012, 2606, 3410, 4462, 5808, 7586, 9898, 12884, 16774, 21890, 28528, 37158, 48410, 63138, 82350, 107312, 139984, 182376, 237746, 310036, 403966, 526646, 686646

Offset: 1

Jeffrey Shallit

Row lengths of A034002 and of A220424. - Reinhard Zumkeller, Dec 15 2012

Satisfies a recurrence of order 72. The characteristic polynomial of this recurrence is a degree-72 polynomial that factors as (x-1)*q(x), where q(x) is a degree-71 polynomial. The unique positive real root of q is approximately 1.3036 and is called Conway's constant (A014715), which equals the limiting ratio a(n+1)/a(n). - Nathaniel Johnston, Apr 12 2018 [Corrected by Richard Stanley, Dec 26 2018]

J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Communications, Springer, NY 1987, pp. 173-188.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452-455.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Peter J. C. Moses, Table of n, a(n) for n = 1..3000 (first 71 terms from Zak Seidov)
S. R. Finch, Conway's Constant.
S. R. Finch, Conway's Constant. [From the Wayback Machine]
Christoph Koutschan,Regular Languages and their Generating Functions: The Inverse Problem.
Eric Weisstein's World of Mathematics, Look and Say Sequence.

a005341 = length . a034002_row  -- Reinhard Zumkeller, Dec 15 2012

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[ Length[ F[ n ] ], {n, 1, 51} ]
p = {12, -18, 18, -18, 18, -20, -22, 31, 15, -4, -4, -19, 62, -50, -21, -11, 41, 54, -56, -44, 15, -27, -15, 45, -8, 89, -64, -66, -25, 38, 126, -39, -32, -33, -65, 107, 14, 16, -13, -79, 7, 42, 12, 8, -26, -9, 35, -23, -20, -30, 34, 58, -1, -20, -36, -6, 13, 8, 6, 3, -1, -4, -1, -4, -5, -1, 8, 6, 0, -6, -4, 1, 0, 1, 1, 1, 1, -1, -1}; q = {-6, 9, -9, 18, -16, 11, -14, 8, -1, 5, -7, -2, -8, 14, 5, 5, -19, -3, 6, 7, 6, -16, 7, -8, 22, -17, 12, -7, -5, -7, 8, -4, 7, 9, -13, 4, 6, -14, 14, -19, 7, 13, -2, 4, -18, 0, 1, 4, 12, -8, 5, 0, -8, -1, -7, 8, 5, 2, -3, -3, 0, 0, 0, 0, 2, 1, 0, -3, -1, 1, 1, 1, -1}; gf = Fold[x #1 + #2 &, 0, p]/Fold[x #1 + #2 &, 0, q]; CoefficientList[Series[gf, {x, 0, 99}], x] (* Peter J. C. Moses, Jun 23 2013 *)

print1(a=1);for(i=2,100,print1(",",#Str(a=A005150(2,a))))  \\ M. F. Hasler, Nov 08 2011

a(n) = A055642(A005150(n)) = A055642(A007651(n)). - Reinhard Zumkeller, Dec 15 2012

More terms from Mike Keith

A034002 A005150 expanded into single digits.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 2, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 3, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 2, 2, 2, 1

Offset: 1

N. J. A. Sloane

			Initial rows                             A005150
  1:  1                                           1
  2:  1,1                                        11
  3:  2,1                                        21
  4:  1,2,1,1                                  1211
  5:  1,1,1,2,2,1                            111221
  6:  3,1,2,2,1,1                            312211
  7:  1,3,1,1,2,2,2,1                      13112221
  8:  1,1,1,3,2,1,3,2,1,1                1113213211
  9:  3,1,1,3,1,2,1,1,1,3,1,2,2,1    31131211131221

Reinhard Zumkeller, Rows n = 1..25 of triangle, flattened
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. DOI: 10.1007/978-1-4612-4808-8_53.
M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 36.
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

See the entry for A005150 for much more about this sequence.
Cf. A088203.
Cf. A005341 (row lengths), A220424 (method B version).

-- see Watkins link, p. 3.
import Data.List (group)
a034002 n k = a034002_tabf !! (n-1) !! (k-1)
a034002_row n = a034002_tabf !! (n-1)
a034002_tabf = iterate
               (concat . map (\xs -> [length xs, head xs]) . group) [1]
-- Reinhard Zumkeller, Aug 09 2012

from sympy import flatten
l=[1]
L=[1]
n=s=1
y=''
while n<21:
    x=str(l[n - 1]) + ' '
    for i in range(len(x) - 1):
        if x[i]==x[i + 1]: s+=1
        else:
            y+=str(s)+str(x[i])
            s=1
    x=''
    n+=1
    l.append(int(y))
    L.append([int(a) for a in list(y)])
    y=''
    s=1
print(l) # A005150
print(flatten(L)) # Indranil Ghosh, Jul 05 2017

A005150(n) = Sum_{k=1..A005341(n)} T(n,k)*10^(A005341(n) - k). - Reinhard Zumkeller, Dec 15 2012

Offset changed and keyword tabf added by Reinhard Zumkeller, Aug 09 2012

cp's OEIS Frontend

A007651 Describe the previous term! (method B - initial term is 1).

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A005341 Length of n-th term in Look and Say sequences A005150 and A007651.

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A034002 A005150 expanded into single digits.

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