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 A106191 #18 Sep 26 2021 05:06:22
%S A106191 1,-1,-3,-7,-17,-45,-129,-393,-1251,-4111,-13835,-47427,-164999,
%T A106191 -581023,-2066823,-7415703,-26805393,-97520733,-356810313,-1312087713,
%U A106191 -4846614093,-17974854933,-66907388973,-249872516253,-935991743553,-3515800038201,-13239692841105
%N A106191 Expansion of sqrt(1-4x)/(1-x).
%C A106191 Row sums of number triangle A106190. Partial sums of A002420.
%C A106191 For n >= 1, the absolute values also give the iterates of A122237, starting from 0. (A122237(0), A122237(A122237(0)), A122237(A122237(A122237(0))), ...), this stems from the fact that the sequence gives the positions of terms with binary expansion 1(10){n-1}0 in A014486 (see A080675).
%F A106191 a(n) = Sum_{k=0..n} binomial(2k, k)/(1-2k).
%F A106191 G.f.: (2/(1-x))/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013
%F A106191 D-finite with recurrence: a(0)=1, a(1)=-1; for n>1, a(n) = (1/n)*((5*n-6)*a(n-1) - (4*n-6)*a(n-2)). - _Tani Akinari_, Aug 25 2013
%Y A106191 |a(n)| = A080300(A080675(n)) = A075161(A001348(n)) (for n >= 1) = A075163(A000244(A008578(n-2))) = A014137(n-1)+A014138(n-2) = 2*A014137(n-1)-1, for n >= 2 (because binomial(2n+2, n+1)/(2n+1) = 2*A000108(n)).
%K A106191 easy,sign
%O A106191 0,3
%A A106191 _Paul Barry_, Apr 24 2005
%E A106191 Barry's formula made more succinct, as well as comments regarding interpretation as absolute values added by _Antti Karttunen_, Sep 14 2006