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A005341 Length of n-th 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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]

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. 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).

LINKS

Zak Seidov and 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

Christoph Koutschan,Regular Languages and their Generating Functions: The Inverse Problem

Eric Weisstein's World of Mathematics, Look and Say Sequence

FORMULA

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

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)

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

CROSSREFS

Sequence in context: A178883 A109832 A039731 * A137268 A008130 A055388

Adjacent sequences:  A005338 A005339 A005340 * A005342 A005343 A005344

KEYWORD

nonn,base,easy,nice,changed

AUTHOR

Jeffrey Shallit

EXTENSIONS

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

STATUS

approved

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Last modified January 20 12:32 EST 2019. Contains 319330 sequences. (Running on oeis4.)