A390560 Upper (1/2)-midsequence of (2^n) and F(n), where F = A000045 (Fibonacci numbers); see Comments.

1, 2, 3, 5, 10, 19, 36, 71, 139, 273, 540, 1069, 2120, 4213, 8381, 16689, 33262, 66335, 132364, 264235, 527671, 1054049, 2106008, 4208633, 8411792, 16814729, 33615129, 67207073, 134376634, 268692571, 537286932, 1074414959, 2148572803, 4296729585, 8592786036

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..34.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-3,1,2).

Formula

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).

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).

Example

s = A000079 = (1, 2, 4, 8, 16, 32, 64, ...).
t = A000045 = (0, 1, 1, 2, 3, 5, 8, 13, ...).
u(n) = (0, 1, 2, 5, 9, 18, 36, 70, 138, ...).
v(n) = (1, 3, 5, 10, 19, 36, 71, 139,, ...).

Mathematica

s[n_] := 2^n ; t[n_] := Fibonacci[n]; r = 1/2;
u[n_] := Floor[r*(s[n] + t[n])]
v[n_] := Ceiling[r*(s[n] + t[n])]
Table[u[n], {n, 0, 60}]  (* A390559 *)
Table[v[n], {n, 0, 60}]  (* A390560 *)

Crossrefs

Cf. A000045, A000079, A390559.

Sequence in context: A078715 A166874 A384177 * A046630 A177874 A293353

Adjacent sequences: A390557 A390558 A390559 * A390561 A390562 A390563

Keyword

nonn,easy

Author

Clark Kimberling, Nov 11 2025

Status

approved