A390349 Upper (1/2)-midsequence of Fibonacci numbers (A000045) and tribonacci numbers (A000213); see Comments.

1, 1, 1, 3, 4, 7, 13, 22, 39, 70, 124, 222, 399, 717, 1293, 2337, 4230, 7671, 13933, 25340, 46145, 84126, 153520, 280408, 512581, 937653, 1716313, 3143363, 5759816, 10558795, 19363765, 35523470, 65188915, 119660130, 219699380, 403459270, 741056491

Offset

0,4

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..36.

Index entries for linear recurrences with constant coefficients, signature (2,1,0,-4,-2,1,2,1).

Formula

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)) = (1, 1, 1, 3, 4, 7, 13, 22).

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

Example

s = A000045 = (0, 1, 1, 2, 3, 5, 8, 13, 21, ...).
t = A000213 = (1, 1, 1, 3, 5, 9, 17, 31, 57, ...).
u(n) = floor((1/2)*(1, 2, 2, 5, 8, 14, 25, 44, 78, ...)) = (0, 1, 1, 2, 4, 7, 12, 22, 39, ...).
v(n) = ceiling((1/2)*(1, 2, 2, 5, 8, 14, 25, 44, 78, ...)) = (1, 1, 1, 3, 4, 7, 13, 22, 39, ...).

Mathematica

s[n_] := Fibonacci[n] ;
t[n_] := LinearRecurrence[{1, 1, 1}, {1, 1, 1}, 50][[n + 1]]; r = 1/2;
u[n_] := Floor[r*(s[n] + t[n])]
v[n_] := Ceiling[r*(s[n] + t[n])]
Table[u[n], {n, 0, 40}]   (* A390348 *)
Table[v[n], {n, 0, 40}]   (* A390349 *)

Crossrefs

Cf. A000045, A000213, A390348.

Sequence in context: A310008 A299024 A116201 * A280224 A378235 A282718

Adjacent sequences: A390346 A390347 A390348 * A390350 A390351 A390352

Keyword

nonn,easy

Author

Clark Kimberling, Nov 10 2025

Status

approved