A355325 Upper midsequence of the Fibonacci numbers (1,2,3,5,8,...) and Lucas numbers (1,3,4,7,11,...); see Comments.

1, 3, 4, 6, 10, 16, 25, 41, 66, 106, 172, 278, 449, 727, 1176, 1902, 3078, 4980, 8057, 13037, 21094, 34130, 55224, 89354, 144577, 233931, 378508, 612438, 990946, 1603384, 2594329, 4197713, 6792042, 10989754, 17781796, 28771550, 46553345, 75324895, 121878240

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) = ceiling((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 + 2 x - 2 x^3 - 2 x^4)/(1 - x - x^2 - x^3 + x^4 + x^5).

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

a(n) = (10 + 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))/30. - Stefano Spezia, Jul 17 2022

Example

a(0) = 1 = ceiling((1+1)/2);
a(1) = 3 = ceiling((2+3)/2);
a(2) = 4 = ceiling((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 *)

Crossrefs

Cf. A000032, A000045, A116470, A355324.

Sequence in context: A310004 A214289 A310005 * A068922 A032408 A347567

Adjacent sequences: A355322 A355323 A355324 * A355326 A355327 A355328

Keyword

nonn,easy

Author

Clark Kimberling, Jul 16 2022

Status

approved