cp's OEIS Frontend

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

%I A112627 #37 Jan 20 2024 09:15:41
%S A112627 1,3,19,51,307,819,4915,13107,78643,209715,1258291,3355443,20132659,
%T A112627 53687091,322122547,858993459,5153960755,13743895347,82463372083,
%U A112627 219902325555,1319413953331,3518437208883,21110623253299,56294995342131,337769972052787,900719925474099
%N A112627 Decimal equivalent of number defined by last k bits of the infinite binary string ...0011001100110011 (numbers with leading zeros omitted).
%C A112627 A182512 is a bisection. - _Olena Kachko_, Dec 16 2023
%H A112627 Harvey P. Dale, <a href="/A112627/b112627.txt">Table of n, a(n) for n = 1..1000</a>
%H A112627 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,16,-16).
%F A112627 G.f.: x*(1+2*x)/(1-x-16*x^2+16*x^3).
%F A112627 a(n) = 4^(n-1) - (4 + (-4)^n)/20. - _Robert Israel_, Sep 02 2014
%F A112627 a(n) = a(n-1)+16*a(n-2)-16*a(n-3) for n>3. - _Colin Barker_, May 19 2016
%e A112627 1 = 1
%e A112627 11 = 3
%e A112627 10011 = 19
%e A112627 110011 = 51
%e A112627 100110011 = 307
%e A112627 1100110011 = 819
%e A112627 ...
%p A112627 seq(4^(n-1) - (4 + (-4)^n)/20, n=1..100); # _Robert Israel_, Sep 02 2014
%t A112627 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 A112627 (* 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 A112627 Table[FromDigits[PadLeft[{},n,{0,0,1,1}],2],{n,60}]//Union (* _Harvey P. Dale_, Mar 15 2016 *)
%o A112627 (PARI) Vec(x*(1+2*x)/((1-x)*(1-4*x)*(1+4*x)) + O(x^50)) \\ _Colin Barker_, May 19 2016
%Y A112627 Cf. A182512.
%K A112627 nonn,base,easy
%O A112627 1,2
%A A112627 _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