A390560 - OEIS

%I #6 Nov 18 2025 21:37:13

%S 1,2,3,5,10,19,36,71,139,273,540,1069,2120,4213,8381,16689,33262,

%T 66335,132364,264235,527671,1054049,2106008,4208633,8411792,16814729,

%U 33615129,67207073,134376634,268692571,537286932,1074414959,2148572803,4296729585,8592786036

%N Upper (1/2)-midsequence of (2^n) and F(n), 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_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1,-3,1,2).

%F a(n) = 3*a(n-1) - a(n-2) - a(n-3) - 3*a(n-4) + a(n-5) + 2*a(n-6) for n>=7, with (a(0),...,a(6)) = (1, 2, 3, 5, 10, 19, 36).

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

%e s = A000079 = (1, 2, 4, 8, 16, 32, 64, ...).

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

%e u(n) = (0, 1, 2, 5, 9, 18, 36, 70, 138, ...).

%e v(n) = (1, 3, 5, 10, 19, 36, 71, 139,, ...).

%t s[n_] := 2^n ; t[n_] := Fibonacci[n]; 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, 60}]  (* A390559 *)

%t Table[v[n], {n, 0, 60}]  (* A390560 *)

%Y Cf. A000045, A000079, A390559.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Nov 11 2025