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 A075193 #39 Sep 08 2022 08:45:07
%S A075193 1,-3,4,-7,11,-18,29,-47,76,-123,199,-322,521,-843,1364,-2207,3571,
%T A075193 -5778,9349,-15127,24476,-39603,64079,-103682,167761,-271443,439204,
%U A075193 -710647,1149851,-1860498,3010349,-4870847,7881196,-12752043,20633239,-33385282,54018521,-87403803,141422324
%N A075193 Expansion of (1-2*x)/(1+x-x^2).
%C A075193 "Inverted" Lucas numbers:
%C A075193 The g.f. is obtained inserting 1/x into the g.f. of Lucas sequence and dividing by x. The closed form is a(n)=(-1)^n*a^(n+1)+(-1)^n*b^(n+1), where a=golden ratio and b=1-a, so that a(n)=(-1)^n*L(n+1), L(n)=Lucas numbers.
%H A075193 Alois P. Heinz, <a href="/A075193/b075193.txt">Table of n, a(n) for n = 0..1000</a>
%H A075193 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A075193 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1).
%F A075193 a(n) = -a(n-1)+a(n-2), a(0)=1, a(1)=-3.
%F A075193 a(n) = term (1,1) in the 1x2 matrix [1,-2] * [-1,1; 1,0]^n. - _Alois P. Heinz_, Jul 31 2008
%F A075193 a(n) = A186679(n)+A186679(n-2) for n>1. - _Reinhard Zumkeller_, Feb 25 2011
%F A075193 a(n) = A039834(n+1)-2*A039834(n). - _R. J. Mathar_, Sep 27 2014
%F A075193 a(n) = (-1)^(n-1)*A001906(n)/A000045(n). - _Taras Goy_, Jan 12 2020
%F A075193 E.g.f.: exp(-(1+sqrt(5))*x/2)*(3 + sqrt(5) - 2*exp(sqrt(5)*x))/(1 + sqrt(5)). - _Stefano Spezia_, Jan 12 2020
%p A075193 a:= n-> (Matrix([[1, -2]]). Matrix([[-1, 1], [1, 0]])^(n))[1, 1]:
%p A075193 seq(a(n), n=0..45); # _Alois P. Heinz_, Jul 31 2008
%t A075193 CoefficientList[Series[(1 - 2z)/(1 + z - z^2), {z, 0, 40}], z]
%o A075193 (Haskell)
%o A075193 a075193 n = a075193_list !! n
%o A075193 a075193_list = 1 : -3 : zipWith (-) a075193_list (tail a075193_list)
%o A075193 -- _Reinhard Zumkeller_, Sep 15 2015
%o A075193 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-2*x)/(1+x-x^2))); // _Marius A. Burtea_, Jan 12 2020
%Y A075193 Cf. A000032, A000045, A001906, A039834, A186679.
%K A075193 easy,sign
%O A075193 0,2
%A A075193 Mario Catalani (mario.catalani(AT)unito.it), Sep 07 2002