A355324 Lower midsequence of the Fibonacci numbers (1,2,3,5,8,...) and Lucas numbers (1,3,4,7,11,...); see Comments.

1, 2, 3, 6, 9, 15, 25, 40, 65, 106, 171, 277, 449, 726, 1175, 1902, 3077, 4979, 8057, 13036, 21093, 34130, 55223, 89353, 144577, 233930, 378507, 612438, 990945, 1603383, 2594329, 4197712, 6792041, 10989754, 17781795, 28771549, 46553345, 75324894, 121878239

Offset

0,2

Comments

Suppose that s = (s(n)) and t = (t(n)) are integer sequences. The lower midsequence, m = m(s,t), of s and t is defined by m(n) = floor((s(n) + t(n))/2). The upper midsequence, M = M(s,t), is defined by M(n) = ceiling((s(n) + t(n))/2).

Here, s(n) = F(n+2) and t(n) = L(n+1), for n >= 0, where F = A000045 (Fibonacci numbers) and L = A000032 (Lucas numbers).

Links

Table of n, a(n) for n=0..38.

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

Formula

a(n) = floor((A000045(n+2) + A000032(n+1))/2).

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) for n >= 5.

G.f.: (1 + x - x^4)/(1 - x - x^2 - x^3 + x^4 + x^5).

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

a(n) = (3*((5 - 4*sqrt(5))*(1 - sqrt(5))^n + (1 + sqrt(5))^n*(5 + 4*sqrt(5)))/2^n + 10*(cos(2*n*Pi/3) - 1))/30. - Stefano Spezia, Jul 17 2022

Example

a(0) = 1 = floor((1+1)/2);
a(1) = 2 = floor((2+3)/2);
a(2) = 3 = floor((3+4)/2).
The Fibonacci and Lucas numbers are interspersed:
1 < 2 < 3 < 4 < 5 < 7 < 8 < 11 < 13 < 18 < 21 < 29 < ...
The midsequences m and M intersperse the ordered union of the Fibonacci and Lucas sequences, A116470, as indicated by the following table:
   F    m    M    L
   1    1    1    1
   2    2    3    3
   3    3    4    4
   5    6    6    7
   8    9   10   11
  13   15   16   18
  21   25   25   29

Mathematica

Table[Floor[(LucasL[n + 1] + Fibonacci[n + 2])/2], {n, 0, 50}]    (* A355324 *)
Table[Ceiling[(LucasL[n + 1] + Fibonacci[n + 2])/2], {n, 0, 50}]  (* A355325 *)

Prog

(Python)
from sympy import fibonacci, lucas
def A355324(n): return fibonacci(n+2)+lucas(n+1)>>1 # Chai Wah Wu, Aug 08 2022

Crossrefs

Cf. A000032, A000045, A116470, A355325.

Sequence in context: A080239 A364755 A114323 * A293631 A018158 A057928

Adjacent sequences: A355321 A355322 A355323 * A355325 A355326 A355327

Keyword

nonn,easy

Author

Clark Kimberling, Jul 16 2022

Status

approved