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

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

Keywords

• nonn
• base
• easy

Comments

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

Examples

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

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).

Crossrefs

Cf. A182512.

Programs

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 *)

PARI

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

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