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Decimal equivalent of number defined by last k bits of the infinite binary string ...0011001100110011 (numbers with leading zeros omitted).
17

%I #37 Jan 20 2024 09:15:41

%S 1,3,19,51,307,819,4915,13107,78643,209715,1258291,3355443,20132659,

%T 53687091,322122547,858993459,5153960755,13743895347,82463372083,

%U 219902325555,1319413953331,3518437208883,21110623253299,56294995342131,337769972052787,900719925474099

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

%C A182512 is a bisection. - _Olena Kachko_, Dec 16 2023

%H Harvey P. Dale, <a href="/A112627/b112627.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,16,-16).

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

%F a(n) = 4^(n-1) - (4 + (-4)^n)/20. - _Robert Israel_, Sep 02 2014

%F a(n) = a(n-1)+16*a(n-2)-16*a(n-3) for n>3. - _Colin Barker_, May 19 2016

%e 1 = 1

%e 11 = 3

%e 10011 = 19

%e 110011 = 51

%e 100110011 = 307

%e 1100110011 = 819

%e ...

%p seq(4^(n-1) - (4 + (-4)^n)/20, n=1..100); # _Robert Israel_, Sep 02 2014

%t 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

%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. *)

%t Table[FromDigits[PadLeft[{},n,{0,0,1,1}],2],{n,60}]//Union (* _Harvey P. Dale_, Mar 15 2016 *)

%o (PARI) Vec(x*(1+2*x)/((1-x)*(1-4*x)*(1+4*x)) + O(x^50)) \\ _Colin Barker_, May 19 2016

%Y Cf. A182512.

%K nonn,base,easy

%O 1,2

%A _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