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 A380696 #72 Mar 27 2025 06:11:01
%S A380696 1,1,0,1,1,4,1,9,4,25,9,64,25,169,64,441,169,1156,441,3025,1156,7921,
%T A380696 3025,20736,7921,54289,20736,142129,54289,372100,142129,974169,372100,
%U A380696 2550409,974169,6677056,2550409,17480761,6677056,45765225,17480761,119814916
%N A380696 a(n) = A007598(floor(n/2) - (-1)^n).
%C A380696 The Fibonacci spiral is produced by creating a quarter circle of radius 1, then adding successive quarter circles such that the radius of the new quarter circle is the sum of the radii of the previous two quarter circles, and that the circumference of the new quarter circle continues where the previous quarter circle ended. When the center of the first quarter circle is at 0,0 the circumference turns clockwise from -1,0, and terms after n=1 are given signs - + + - repeating, these are the x coordinates where the circumferences meet. The y coordinates are the golden rectangle numbers (A001654) with the same pattern of alternation (x,a,b,x), and the same pattern of signs shifted backward one.
%H A380696 Paolo Xausa, <a href="/A380696/b380696.txt">Table of n, a(n) for n = 0..1000</a>
%H A380696 Benjamin G. Brunsden, <a href="/A380696/a380696_1.png">Sequence shown as coordinates on the Fibonacci spiral</a>
%H A380696 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,2,0,-1).
%F A380696 a(n) = Fibonacci(floor(n/2)-(-1)^n)^2.
%F A380696 a(n) = A053602(n-2)^2 for n >= 2.
%F A380696 a(n) = A272912(n)^2 for n >= 3.
%F A380696 G.f. ( 1+x-x^3-x^4-2*x^2 ) / ( (1+x^2)*(x^2-x-1)*(x^2+x-1) ).
%F A380696 a(2*n) + a(2*n+1) = A069921(n-1) for n>=1.
%t A380696 A380696[n_] := Fibonacci[Floor[n/2] - (-1)^n]^2; Array[A380696, 50, 0] (* or *)
%t A380696 LinearRecurrence[{0, 2, 0, 2, 0, -1}, {1, 1, 0, 1, 1, 4}, 50] (* _Paolo Xausa_, Mar 27 2025 *)
%o A380696 (Python)
%o A380696 from sympy import fibonacci
%o A380696 def A380696(n): return fibonacci(n+1>>1 if n&1 else (n>>1)-1)**2 # _Chai Wah Wu_, Mar 26 2025
%Y A380696 Cf. A000045, A007598, A053602, A069921, A123231, A272912, A001654.
%K A380696 nonn,easy
%O A380696 0,6
%A A380696 _Benjamin G. Brunsden_, Jan 30 2025