A272912 -id:A272912 - cp's OEIS frontend

A272911 Difference sequence of the sequence the increasing sequence of products of two Lucas numbers A000032.

2, 1, 3, 2, 2, 1, 4, 2, 3, 7, 1, 4, 11, 3, 2, 5, 18, 4, 1, 10, 29, 5, 2, 3, 15, 47, 10, 1, 4, 25, 76, 15, 3, 2, 5, 40, 123, 25, 4, 1, 10, 65, 199, 40, 5, 2, 3, 15, 105, 322, 65, 10, 1, 4, 25, 170, 521, 105, 15, 3, 2, 5, 40, 275, 843, 170, 25, 4, 1, 10, 65

Offset: 1

Clark Kimberling, May 10 2016

Conjecture: every term is a product of two Lucas numbers or a product of two Fibonacci numbers.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000032, A000045, A272909, A272912.

z = 100; t = Take[Sort[Flatten[Table[LucasL[m] LucasL[n], {n, 1, z}, {m, n, z}]]],   1000]; Differences[t]

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

Original entry on oeis.org

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

Benjamin G. Brunsden, Jan 30 2025

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.

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).

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

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 *)

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

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.

cp's OEIS Frontend

A272911 Difference sequence of the sequence the increasing sequence of products of two Lucas numbers A000032.

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