

A005341


Length of nth term in Look and Say sequences A005150 and A007651.
(Formerly M0321)


10



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
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OFFSET

1,2


COMMENTS

Satisfies a recurrence of order 72. The characteristic polynomial of this recurrence is a degree72 polynomial that factors as (x1)*q(x), where q(x) is a degree71 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]


REFERENCES

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. 173188.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452455.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA



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[ 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 *)


PROG

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


CROSSREFS



KEYWORD

nonn,base,easy,nice


AUTHOR



EXTENSIONS

More terms from Mike Keith (Domnei(AT)aol.com)


STATUS

approved



