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 A253198 #43 Dec 13 2023 08:51:59
%S A253198 0,1,2,4,5,10,16,25,42,68,109,178,288,465,754,1220,1973,3194,5168,
%T A253198 8361,13530,21892,35421,57314,92736,150049,242786,392836,635621,
%U A253198 1028458,1664080,2692537,4356618,7049156,11405773,18454930,29860704,48315633,78176338,126491972,204668309,331160282,535828592
%N A253198 a(n) = a(n-1) + a(n-2) - (-1)^(a(n-1) + a(n-2)) with a(0)=0, a(1)=1.
%C A253198 This is a minimally modified Fibonacci sequence (A000045) in that it preserves characteristic properties of the original sequence: a(n) is a function of the sum of the preceding two terms, the ratio of two consecutive terms tends to the Golden Mean, and the initial two terms are the same as in the Fibonacci sequence. See A253197 and A255978 for other members of this family.
%H A253198 Colin Barker, <a href="/A253198/b253198.txt">Table of n, a(n) for n = 0..1000</a>
%H A253198 W. Puszkarz, <a href="http://vixra.org/abs/1503.0113">A Note on Minimal Extensions of the Fibonacci Sequence</a>, viXra:1503.0113, 2015.
%H A253198 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,-1).
%F A253198 a(n) = a(n-1) + a(n-2) - (-1)^(a(n-1) + a(n-2)), a(0)=0, a(1)=1.
%F A253198 a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) for n>4. - _Colin Barker_, Mar 28 2015
%F A253198 G.f.: -x*(2*x^3-x^2-x-1) / ((x-1)*(x^2+x-1)*(x^2+x+1)). - _Colin Barker_, Mar 28 2015
%F A253198 a(n) = 2*A000045(n) - A079978(n+2). - _Nicolas Bělohoubek_, Aug 16 2021
%e A253198 For n=2, a(2) = 0 + 1 - (-1)^1 = 0 + 1 + 1 = 2.
%e A253198 For n=3, a(3) = 1 + 2 - (-1)^3 = 1 + 2 + 1 = 4.
%e A253198 For n=4, a(4) = 2 + 4 - (-1)^6 = 2 + 4 - 1 = 5.
%t A253198 RecurrenceTable[{a[n]==a[n-1]+a[n-2] -(-1)^(a[n-1]+a[n-2]), a[0]==0, a[1]==1}, a, {n, 0, 50}]
%t A253198 LinearRecurrence[{1,1,1,-1,-1},{0,1,2,4,5},50] (* _Harvey P. Dale_, Mar 17 2019 *)
%o A253198 (Magma) [n le 2 select (n-1) else Self(n-1) + Self(n-2) - (-1)^(Self(n-1) + Self(n-2)): n in [1..50] ]; // _Vincenzo Librandi_, Mar 28 2015
%o A253198 (PARI) concat(0, Vec(-x*(2*x^3-x^2-x-1)/((x-1)*(x^2+x-1)*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, Mar 28 2015
%Y A253198 Cf. A000045, A253197, A255978, A079978.
%K A253198 nonn,easy
%O A253198 0,3
%A A253198 _Waldemar Puszkarz_, Mar 24 2015