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A179999 Length of the n-th term in the modified Look and Say sequence A110393. 2
1, 2, 2, 4, 6, 8, 10, 14, 18, 24, 30, 40, 50, 66, 82, 108, 134, 176, 218, 286, 354, 464, 574, 752, 930, 1218, 1506, 1972, 2438, 3192, 3946, 5166, 6386, 8360, 10334, 13528, 16722, 21890, 27058, 35420, 43782, 57312, 70842, 92734, 114626, 150048 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The average multiplicative growth from the n-th term to the (n+1)-st term is sqrt(phi) = 1.272..., where phi = (1+sqrt(5))/2 is the golden ratio, see A139339.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

N. Johnston, Further Variants of the “Look-and-Say” Sequence

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1).

FORMULA

a(n) = length(A110393(n)).

From Colin Barker, Aug 10 2019: (Start)

G.f.: x*(1 + x)*(1 + x - x^2)*(1 - x + x^2) / ((1 - x)*(1 - x^2 - x^4)).

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n>6.

(End)

From A.H.M. Smeets, Aug 10 2019 (Start)

Lim_{n -> inf} a(n+1)/a(n) = (1+phi)/2 = (3+sqrt(5)/4 = A239798 for odd n.

Lim_{n -> inf} a(n+1)/a(n) = 2/phi = 4/(1+sqrt(5)) = A134972 for even n.

Lim_{n -> inf} a(n+2)/a(n) = (1+phi)/phi = phi = A001622. (End)

For odd n > 1, a(n) = 4*Fibonacci((n + 1)/2) - 2. For even n, a(n) = 2*Fibonacci(n/2 + 2) - 2. - Ehren Metcalfe, Aug 10 2019

EXAMPLE

The 6th term in A110393 is 21112211, so a(6) = 8.

MATHEMATICA

CoefficientList[Series[((1+x) (-1-x+x^2) (1-x+x^2))/((1-x) (-1+x^2+x^4)), {x, 0, 99}], x] (* Peter J. C. Moses, Jun 23 2013 *)

PROG

(PARI) Vec(x*(1 + x)*(1 + x - x^2)*(1 - x + x^2) / ((1 - x)*(1 - x^2 - x^4)) + O(x^50)) \\ Colin Barker, Aug 10 2019

CROSSREFS

Cf. A005341, A049194, A098596, A110393.

Cf. A001622, A134972, A139339, A239798.

Sequence in context: A145809 A309711 A116859 * A286736 A241383 A258125

Adjacent sequences:  A179996 A179997 A179998 * A180000 A180001 A180002

KEYWORD

nonn,base,easy

AUTHOR

Nathaniel Johnston, Jan 13 2011

STATUS

approved

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Last modified January 19 00:40 EST 2020. Contains 331030 sequences. (Running on oeis4.)