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A179999
Length of the n-th term in the modified Look and Say sequence A110393.
3
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.
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
KEYWORD
nonn,base,easy
AUTHOR
Nathaniel Johnston, Jan 13 2011
STATUS
approved