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 A167193 #20 Sep 08 2022 08:45:48
%S A167193 1,0,0,2,-4,10,-20,42,-84,170,-340,682,-1364,2730,-5460,10922,-21844,
%T A167193 43690,-87380,174762,-349524,699050,-1398100,2796202,-5592404,
%U A167193 11184810,-22369620,44739242,-89478484,178956970,-357913940,715827882,-1431655764,2863311530,-5726623060,11453246122,-22906492244,45812984490
%N A167193 a(n) = (1/3)*(1 - (-2)^n + 3*(-1)^n ) = (-1)^(n+1)*A167030(n).
%C A167193 This is the inverse binomial transform of 1, 1, 1, 3, 5, 11,.. (continued as in A001045 and conjectured to be equal to A152046).
%C A167193 Any sequence (like this one) which obeys a(n)= -2a(n-1)+a(n-2)+2a(n-3) also obeys a(n)=5a(n-2)-4a(n-4), proved by telescoping; see A101622.
%H A167193 Vincenzo Librandi, <a href="/A167193/b167193.txt">Table of n, a(n) for n = 0..1000</a>
%H A167193 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (-2,1,2).
%F A167193 a(2n) = (-1)^n* A084240(n). a(2n+1) = A020988(n).
%F A167193 G.f.: ( -1 - 2*x + x^2 ) / ( (x-1)*(1+2*x)*(1+x) ).
%F A167193 a(n) = -a(n-1) + 2*a(n-2) - 2*(-1)^n.
%F A167193 a(n) = -2*a(n-1) + a(n-2) + 2*a(n-3).
%F A167193 E.g.f.: (1/3)*(exp(x) + 3*exp(-x) - exp(-2*x)). - _G. C. Greubel_, Jun 04 2016
%t A167193 LinearRecurrence[{-2,1,2},{1,0,0}, 25] (* or *) Table[(1/3)*(1 + 3*(-1)^n - (-2)^n), {n,0,25}] (* _G. C. Greubel_, Jun 04 2016 *)
%o A167193 (Magma) [( 1-(-1)^n*2^n)/3+(-1)^n: n in [0..40] ]; // _Vincenzo Librandi_, Aug 06 2011
%K A167193 easy,sign
%O A167193 0,4
%A A167193 _Paul Curtz_, Oct 30 2009