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

Author

Nathaniel Johnston, Jan 13 2011

Keywords

• nonn
• base
• easy

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.

Examples

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

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).

Crossrefs

Cf. A005341, A049194, A098596, A110393.

Cf. A001622, A134972, A139339, A239798.

Programs

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 *)

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

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