0,1

Suppose that s = (s(n)) and t = (t(n)) are sequences of numbers and r > 0. The lower (r)-midsequence of s and t is given by u = floor(r*(s + t)); the upper r-midsequence of s and t is given by v = ceiling(r*(s + t)). If s and t are linearly recurrent and r is rational, then u and v are linearly recurrent.

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

a(n) = ceiling((2/3)*(A000032(n) + A000045(n))).

a(n) = a(n-1) + a(n-2) + a(n-8) - a(n-9) - a(n-10).

s = (F(n)) = (0, 1, 1, 2, 3, 5, 8, ...); t = (2, 1, 3, 4, 7, 11, 18, ...).

u(n) = floor((2/3)(0+2, 1+1, 1+3, 2+4, 3+7, ...)) = (1, 1, 2, 4, 6, ...).

v(n) = ceiling((2/3)(0+2, 1+1, 1+3, 2+4, 3+7, ...)) = (2, 2, 3, 4, 7, ...).

s[n_] := Fibonacci[n]; t[n_] := LucasL[n]; r = 2/3;

u[n_] := Floor[r*(s[n] + t[n])]

v[n_] := Ceiling[r*(s[n] + t[n])]

Table[u[n], {n, 0, 30}] (* A389121 *)

Table[v[n], {n, 0, 30}] (* A389122 *)

(Python)

from sympy import lucas, fibonacci

def A389122(n): return ((lucas(n)+fibonacci(n)<<1)-1)//3+1 # Chai Wah Wu, Oct 08 2025

Cf. A000032, A000045, A389121.

Sequence in context: A110871 A243856 A173433 * A053638 A051920 A382247

Adjacent sequences: A389119 A389120 A389121 * A389123 A389124 A389125

nonn,easy

Clark Kimberling, Oct 03 2025

approved