A387778 - OEIS

%I #12 Sep 26 2025 09:04:48

%S 0,1,4,10,27,72,188,493,1292,3382,8855,23184,60696,158905,416020,

%T 1089154,2851443,7465176,19544084,51167077,133957148,350704366,

%U 918155951,2403763488,6293134512,16475640049,43133785636,112925716858,295643364939,774004377960

%N Lower (1/2)-midsequence of (F(2n)) and (F(2n+1)), 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 (3,-1,1,-3,1).

%F a(n) = floor((1/2)*(F(2*n) + F(2*n+1))), where F=A000045.

%F a(n) = 3*a(n-1) - a(n-2) + a(n-3) - 3*a(n-4) + a(n-5).

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

%F 6* a(n) = 3*A001906(n+1) -A061347(n+1) -2 ; - _R. J. Mathar_, Sep 26 2025

%e s = (F(2n)) = (0, 1, 3, 8, 21, ...); t = (1, 2, 5, 13, 34, ...).

%e u(n) = floor((1/2)(0+1, 1+2, 3+5, 8+13, 21+34, ...)) = (0, 1, 4, 10, 27, ...).

%e v(n) = ceiling((1/2)(0+1, 1+2, 3+5, 8+13, 21+34, ...)) = (1, 2, 4, 11, 28, ...).

%t f[n_] := Fibonacci[n]; s[n_] := f[2 n]; t[n_] := f[2 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, 30}]  (* lower, A387778 *)

%t Table[v[n], {n, 0, 30}]  (* upper, A387779 *)

%t (* Also *)

%t LinearRecurrence[{3, -1, 1, -3, 1}, {0, 1, 4, 10, 27}, 30]

%Y Cf. A000045, A387779, A387780, A387781.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Sep 08 2025