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 A258085 #28 Sep 08 2022 08:46:12
%S A258085 0,1,2,3,4,5,7,8,12,13,20,21,33,34,54,55,88,89,143,144,232,233,376,
%T A258085 377,609,610,986,987,1596,1597,2583,2584,4180,4181,6764,6765,10945,
%U A258085 10946,17710,17711,28656,28657,46367,46368,75024,75025,121392,121393,196417
%N A258085 Strictly increasing list of F and F - 1, where F = A000045, the Fibonacci numbers.
%C A258085 Beginning with a(4) = 3, these are the numbers m such that if r = golden ratio and the fractional parts {r}, {2 r}, ..., {mr} are arranged in increasing order, then the set of differences {kr} - {(k - 1)r}, for k = 2..m, consists of exactly two numbers.
%H A258085 Clark Kimberling, <a href="/A258085/b258085.txt">Table of n, a(n) for n = 1..1000</a>
%H A258085 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,0,0,-1).
%F A258085 From _Colin Barker_, Jun 28 2015: (Start)
%F A258085 a(n) = 2*a(n-2) - a(n-6) for n>8.
%F A258085 G.f.: -x^2*(x^6+x^5+x^4-x^2-2*x-1) / ((x-1)*(x+1)*(x^4+x^2-1)).
%F A258085 (End)
%F A258085 For n >= 3, a(n) = A000045(A004526(n+5)) - A000035(n). - _Robert Israel_, Jun 29 2015
%e A258085 F = (1,1,2,3,5,8,13,...); F-1 = (0,0,1,2,4,7,12,...), so that the ordered list of F and F-1 is (0,1,2,3,4,5,7,8,...).
%e A258085 Regarding the fractional parts in Comment, for r = golden ratio and m = 7, the fractional parts are ordered as follows: -8+r, -3+2r, -11+7r, -6+4 r,-1+r, -9+6r, -4+3r. The set of differences is {5-3r, -8+5r}, so that 7 is a term in A258085.
%p A258085 map((t->(t-1,t)) @ combinat:-fibonacci,[1,$4..100]); # _Robert Israel_, Jun 29 2015
%t A258085 f = Fibonacci[Range[60]]; u = Union[f, f - 1]
%o A258085 (Magma) [0,1] cat &cat[[Fibonacci(n)-1, Fibonacci(n)]: n in [4..40]]; // _Vincenzo Librandi_, Jun 28 2015
%o A258085 (PARI) a(n)=if(n<6,n-1,fibonacci((n+5)\2)-n%2) \\ _Charles R Greathouse IV_, Jun 28 2015
%o A258085 (PARI) concat(0, Vec(-x^2*(x^6+x^5+x^4-x^2-2*x-1)/((x-1)*(x+1)*(x^4+x^2-1)) + O(x^100))) \\ _Colin Barker_, Feb 16 2017
%Y A258085 Cf. A000035, A000045, A004526.
%K A258085 nonn,easy
%O A258085 1,3
%A A258085 _Clark Kimberling_, Jun 27 2015