A380696 a(n) = A007598(floor(n/2) - (-1)^n).

1, 1, 0, 1, 1, 4, 1, 9, 4, 25, 9, 64, 25, 169, 64, 441, 169, 1156, 441, 3025, 1156, 7921, 3025, 20736, 7921, 54289, 20736, 142129, 54289, 372100, 142129, 974169, 372100, 2550409, 974169, 6677056, 2550409, 17480761, 6677056, 45765225, 17480761, 119814916

Offset

0,6

Comments

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.

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Benjamin G. Brunsden, Sequence shown as coordinates on the Fibonacci spiral

Index entries for linear recurrences with constant coefficients, signature (0,2,0,2,0,-1).

Formula

a(n) = Fibonacci(floor(n/2)-(-1)^n)^2.

a(n) = A053602(n-2)^2 for n >= 2.

a(n) = A272912(n)^2 for n >= 3.

G.f. ( 1+x-x^3-x^4-2*x^2 ) / ( (1+x^2)*(x^2-x-1)*(x^2+x-1) ).

a(2*n) + a(2*n+1) = A069921(n-1) for n>=1.

Mathematica

A380696[n_] := Fibonacci[Floor[n/2] - (-1)^n]^2; Array[A380696, 50, 0] (* or *)
LinearRecurrence[{0, 2, 0, 2, 0, -1}, {1, 1, 0, 1, 1, 4}, 50] (* Paolo Xausa, Mar 27 2025 *)

Prog

(Python)
from sympy import fibonacci
def A380696(n): return fibonacci(n+1>>1 if n&1 else (n>>1)-1)**2 # Chai Wah Wu, Mar 26 2025

Crossrefs

Cf. A000045, A007598, A053602, A069921, A123231, A272912, A001654.

Sequence in context: A175643 A143864 A296483 * A073364 A125165 A259448

Adjacent sequences: A380693 A380694 A380695 * A380697 A380698 A380699

Keyword

nonn,easy

Author

Benjamin G. Brunsden, Jan 30 2025

Status

approved