A112627 Decimal equivalent of number defined by last k bits of the infinite binary string ...0011001100110011 (numbers with leading zeros omitted).

1, 3, 19, 51, 307, 819, 4915, 13107, 78643, 209715, 1258291, 3355443, 20132659, 53687091, 322122547, 858993459, 5153960755, 13743895347, 82463372083, 219902325555, 1319413953331, 3518437208883, 21110623253299, 56294995342131, 337769972052787, 900719925474099

Offset

1,2

Comments

A182512 is a bisection. - Olena Kachko, Dec 16 2023

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,16,-16).

Formula

G.f.: x*(1+2*x)/(1-x-16*x^2+16*x^3).

a(n) = 4^(n-1) - (4 + (-4)^n)/20. - Robert Israel, Sep 02 2014

a(n) = a(n-1)+16*a(n-2)-16*a(n-3) for n>3. - Colin Barker, May 19 2016

Example

1 = 1
11 = 3
10011 = 19
110011 = 51
100110011 = 307
1100110011 = 819
...

Maple

seq(4^(n-1) - (4 + (-4)^n)/20, n=1..100); # Robert Israel, Sep 02 2014

Mathematica

t = {}; lst = First@RealDigits[ N[1/5, 100], 2]; Do[ If[ lst[[n]] == 1, AppendTo[t, FromDigits[ Reverse@Take[lst, n], 2]]], {n, 49}]; t
(* The first line establishes the binary expansion of 1/5 to 100 places (A021913, except for start). The loop extracts the first n terms in this sequence and if it ends in "1", reverses digits and converts to decimal. *)
Table[FromDigits[PadLeft[{}, n, {0, 0, 1, 1}], 2], {n, 60}]//Union (* Harvey P. Dale, Mar 15 2016 *)

Prog

(PARI) Vec(x*(1+2*x)/((1-x)*(1-4*x)*(1+4*x)) + O(x^50)) \\ Colin Barker, May 19 2016

Crossrefs

Cf. A182512.

Sequence in context: A214883 A363697 A239449 * A222185 A265774 A100697

Adjacent sequences: A112624 A112625 A112626 * A112628 A112629 A112630

Keyword

nonn,base,easy

Author

N. J. A. Sloane, based on email from Artur Jasinski, with assistance from Dean Hickerson, Ray Chandler and Robert G. Wilson v, Dec 27 2005

Status

approved