A390350 - OEIS

%I #14 Dec 28 2025 09:57:39

%S 1,3,4,7,12,19,31,51,82,133,216,349,565,915,1480,2395,3876,6271,10147,

%T 16419,26566,42985,69552,112537,182089,294627,476716,771343,1248060,

%U 2019403,3267463,5286867,8554330,13841197,22395528,36236725,58632253,94868979,153501232

%N Lower (1/2)-midsequence of F(n) and F(n+4), where F = A000045 (Fibonacci numbers); 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_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,-1).

%F a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5), with (a(0),...,a(4)) = (1, 3, 4, 7, 12).

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

%F E.g.f.: exp(-x/2)*(-5*(2*exp(3*x/2) + cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + 9*exp(x)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2)))/30. - _Stefano Spezia_, Dec 27 2025

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

%e (t(n)) = A000045(n+4) = (3, 5, 8, 13, 21, 34, 55, ...).

%e u(n) = (1, 3, 4, 7, 12, 19, 31, 51, 82, ...).

%e v(n) = (2, 3, 5, 8, 12, 20, 32, 51, 83, ...).

%t s[n_] := Fibonacci[n] ; t[n_] := Fibonacci[n + 4]; 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}]   (* A390350 *)

%t Table[v[n], {n, 0, 40}]   (* A390351 *)

%Y Cf. A000045, A390351.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Nov 11 2025