A270743 Runlength sequence of the zero-one sequence A270742.

1, 1, 2, 1, 2, 4, 1, 3, 6, 1, 4, 8, 1, 5, 10, 1, 6, 12, 1, 7, 14, 1, 8, 16, 1, 9, 18, 1, 10, 20, 1, 11, 22, 1, 12, 24, 1, 13, 26, 1, 14, 28, 1, 15, 30, 1, 16, 32, 1, 17, 34, 1, 18, 36, 1, 19, 38, 1, 20, 40, 1, 21, 42, 1, 22, 44, 1, 23, 46, 1, 24, 48, 1, 25

Offset

1,3

Comments

This is the runlength sequence for C = (1/2)*(3/4)*(7/8)*(15/16)... in each base (2,3,4,...); for base 10, we have C = 0.8900100999989990000001000..., as in A132038.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Formula

a(3n) = 2n for n > 0, a(3n+1) = 1 for n >= 0, a(3n+2) = n + 1 for n >= 0.

G.f.: x*(1 + x + 2*x^2 - x^3)/((1 - x)*(1 + x + x^2)). - Andrew Howroyd, Oct 31 2025

Example

(zero-one sequence at A270742) = (0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,...), with first 6 runs 0, 1, 00, 1, 00, 1111 of lengths 1,1,2,1,2,4.

Mathematica

d = Join[{0}, RealDigits[QPochhammer[1/2], 2, 200000][[1]]];
u = Map[Length, Split[d]]

Crossrefs

Cf. A132038, A270742.

Sequence in context: A165092 A306915 A380890 * A209750 A156042 A287691

Adjacent sequences: A270740 A270741 A270742 * A270744 A270745 A270746

Keyword

nonn,easy

Author

Clark Kimberling, Apr 04 2016

Status

approved