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 A253197 #37 Sep 08 2022 08:46:10
%S A253197 0,1,3,4,9,15,24,41,67,108,177,287,464,753,1219,1972,3193,5167,8360,
%T A253197 13529,21891,35420,57313,92735,150048,242785,392835,635620,1028457,
%U A253197 1664079,2692536,4356617,7049155,11405772,18454929,29860703,48315632,78176337,126491971,204668308,331160281,535828591,866988872
%N A253197 a(n) = a(n-1) + a(n-2) + (1 - (-1)^(a(n-1) + a(n-2))) with a(0) = 0, a(1) = 1.
%C A253197 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 A253198 and A255978 for other members of this family.
%H A253197 Colin Barker, <a href="/A253197/b253197.txt">Table of n, a(n) for n = 0..1000</a>
%H A253197 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 A253197 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,-1).
%F A253197 a(n) = a(n-1) + a(n-2) + (1 - (-1)^(a(n-1) + a(n-2))), a(0) = 0, a(1) = 1.
%F A253197 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 A253197 G.f.: x*(x^3 + 2*x + 1) / ((x-1)*(x^2 + x - 1)*(x^2 + x + 1)). - _Colin Barker_, Mar 28 2015
%F A253197 a(n) = 2*Fibonacci(n+1) - (1 if n == 0 (mod 3)) - 1. - _Nicolas Bělohoubek_, Sep 29 2021
%e A253197 For n = 2, a(2) = 0 + 1 + (1 - (-1)^1) = 0 + 1 + 2 = 3.
%e A253197 For n = 3, a(3) = 1 + 3 + (1 - (-1)^4) = 1 + 3 + 0 = 4.
%e A253197 For n = 4, a(4) = 3 + 4 + (1 - (-1)^7) = 3 + 4 + 2 = 9.
%t A253197 RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] + (1 - (-1)^(a[n - 1] + a[n - 2])), a[0] == 0, a[1] == 1}, a, {n, 0, 50}]
%o A253197 (Magma) [n le 2 select (n-1) else Self(n-1) + Self(n-2) + (1 - (-1)^(Self(n-1) + Self(n-2))): n in [1..50] ]; // _Vincenzo Librandi_, Mar 28 2015
%o A253197 (PARI) concat(0, Vec(x*(x^3+2*x+1)/((x-1)*(x^2+x-1)*(x^2+x+1)) + O(x^100))) \\ _Colin Barker_, Mar 28 2015
%Y A253197 Cf. A000045, A253198, A255978.
%K A253197 nonn,easy
%O A253197 0,3
%A A253197 _Waldemar Puszkarz_, Mar 24 2015