This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A316742 #51 Sep 16 2018 07:00:34
%S A316742 1,0,3,1,7,3,15,7,31,15,63,31,127,63,255,127,511,255,1023,511,2047,
%T A316742 1023,4095,2047,8191,4095,16383,8191,32767,16383,65535,32767,131071,
%U A316742 65535,262143,131071,524287,262143,1048575,524287,2097151,1048575,4194303,2097151,8388607,4194303
%N A316742 Stepping through the Mersenne sequence (A000225) one step back, two steps forward.
%H A316742 Jim Singh and others, <a href="http://www.mersenneforum.org/showthread.php?t=23501">Fascinating periodic sequence pairs</a>, Mersenne Forum thread, July 2018.
%H A316742 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2).
%F A316742 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n>2, a(0)=1, a(1)=0, a(2)=3.
%F A316742 From _Bruno Berselli_, Jul 12 2018: (Start)
%F A316742 G.f.: (1 - x + x^2)/((1 - x)*(1 - 2*x^2)).
%F A316742 a(n) = 2*a(n-2) + 1 for n>1, a(0)=1, a(1)=0.
%F A316742 a(n) = (1 + (-1)^n)*(2^(n/2) - 2^((n-3)/2)) + 2^((n-1)/2) - 1.
%F A316742 Therefore: a(4*k) = 2*4^k - 1, a(4*k+1) = 4^k - 1, a(4*k+2) = 4^(k+1) - 1, a(4*k+3) = 2*4^k - 1. (End)
%e A316742 Let 1. The first four terms are 1, (1-1)/2 = 0, 2*1+1 = 3, 1.
%e A316742 Let 4*1+3 = 7. The next four terms are 7, (7-1)/2 = 3, 2*7+1 = 15, 7.
%e A316742 Let 4*7+3 = 31. The next four terms are 31, (31-1)/2 = 15, 2*31+1 = 63, 31; etc.
%p A316742 seq(coeff(series((1-x+x^2)/((1-x)*(1-2*x^2)), x,n+1),x,n),n=0..45); # _Muniru A Asiru_, Jul 14 2018
%t A316742 CoefficientList[Series[(1 - x + x^2)/((1 - x) (1 - 2 x^2)), {x, 0, 42}], x] (* _Michael De Vlieger_, Jul 13 2018 *)
%t A316742 LinearRecurrence[{1, 2, -2}, {1, 0, 3}, 46] (* _Robert G. Wilson v_, Jul 21 2018 *)
%o A316742 (GAP) a:=[1,0,3];; for n in [4..45] do a[n]:=a[n-1]+2*a[n-2]-2*a[n-3]; od; a; # _Muniru A Asiru_, Jul 14 2018
%Y A316742 Cf. A000225, A024036, A083420.
%K A316742 nonn,easy,less
%O A316742 0,3
%A A316742 _Jim Singh_, Jul 12 2018