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
Examples
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
Links
- Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1).
Programs
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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 *) -
Python
from sympy import fibonacci, lucas def A355324(n): return fibonacci(n+2)+lucas(n+1)>>1 # Chai Wah Wu, Aug 08 2022
Formula
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
Comments