A179999 Length of the n-th term in the modified Look and Say sequence A110393.

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

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)

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

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

Limit_{n->oo} 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