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%I M0321 #60 Oct 29 2023 21:10:52
%S 1,2,2,4,6,6,8,10,14,20,26,34,46,62,78,102,134,176,226,302,408,528,
%T 678,904,1182,1540,2012,2606,3410,4462,5808,7586,9898,12884,16774,
%U 21890,28528,37158,48410,63138,82350,107312,139984,182376,237746,310036,403966,526646,686646
%N Length of n-th term in Look and Say sequences A005150 and A007651.
%C Row lengths of A034002 and of A220424. - _Reinhard Zumkeller_, Dec 15 2012
%C 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]
%D 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.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452-455.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Peter J. C. Moses, <a href="/A005341/b005341.txt">Table of n, a(n) for n = 1..3000</a> (first 71 terms from Zak Seidov)
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/cnwy/cnwy.html">Conway's Constant</a>.
%H S. R. Finch, <a href="http://web.archive.org/web/20010207194413 /http://www.mathsoft.com/asolve/constant/cnwy/cnwy.html">Conway's Constant</a>. [From the Wayback Machine]
%H Christoph Koutschan,<a href="http://www.risc.jku.at/research/combinat/software/RLangGFun/pub/diplomarbeit.pdf">Regular Languages and their Generating Functions: The Inverse Problem</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LookandSaySequence.html">Look and Say Sequence</a>.
%F a(n) = A055642(A005150(n)) = A055642(A007651(n)). - _Reinhard Zumkeller_, Dec 15 2012
%t 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} ]
%t 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 *)
%o (PARI) print1(a=1);for(i=2,100,print1(",",#Str(a=A005150(2,a)))) \\ _M. F. Hasler_, Nov 08 2011
%o (Haskell)
%o a005341 = length . a034002_row -- _Reinhard Zumkeller_, Dec 15 2012
%K nonn,base,easy,nice
%O 1,2
%A _Jeffrey Shallit_
%E More terms from Mike Keith (Domnei(AT)aol.com)
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