A390348 - OEIS

%I #10 Dec 27 2025 13:59:24

%S 0,1,1,2,4,7,12,22,39,69,124,222,398,717,1293,2336,4230,7671,13932,

%T 25340,46145,84125,153520,280408,512580,937653,1716313,3143362,

%U 5759816,10558795,19363764,35523470,65188915,119660129,219699380,403459270,741056490

%N Lower (1/2)-midsequence of Fibonacci numbers (A000045) and tribonacci numbers (A000213); see Comments.

%C 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.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,0,-4,-2,1,2,1).

%F a(n) = 2*a(n-1) + a(n-2) - 4*a(n-4) - 2*a(n-5) + a(n-6) + 2*a(n-7) + a(n-8), with (a(0),...,a(7)) = (0, 1, 1, 2, 4, 7, 12, 22).

%F G.f.: x*(-1 + x + x^2 + x^3 - x^4)/(-1 + 2*x + x^2 - 4*x^4 - 2*x^5 + x^6 + 2*x^7 + x^8).

%e s = A000045 = (0, 1, 1, 2, 3, 5, 8, 13, 21, ...).

%e t = A000213 = (1, 1, 1, 3, 5, 9, 17, 31, 57, ...).

%e u(n) = floor((1/2)*(1, 2, 2, 5, 8, 14, 25, 44, 78, ...)) = (0, 1, 1, 2, 4, 7, 12, 22, 39, ...).

%e v(n) = ceiling((1/2)*(1, 2, 2, 5, 8, 14, 25, 44, 78, ...)) = (1, 1, 1, 3, 4, 7, 13, 22, 39, ...).

%t s[n_] := Fibonacci[n] ;

%t t[n_] := LinearRecurrence[{1, 1, 1}, {1, 1, 1}, 50][[n + 1]]; r = 1/2;

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

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

%t Table[u[n], {n, 0, 40}]   (* A390348 *)

%t Table[v[n], {n, 0, 40}]   (* A390349 *)

%Y Cf. A000045, A000213, A390349.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Nov 10 2025