0,3

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.

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

a(n) = ceiling((3*n^2 + n)/6).

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

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

s = (n^2) = A000290 = (0, 1, 4, 9, 16, 25, ...).

t = (n*(n+1)/2) = A000217 = (0, 1, 3, 6, 10, 15, ...).

u(n) = floor((1/3)(0+0, 1+1, 4+3, 9+6, 16+10, ...)) = (0, 0, 2, 5, 8, ...).

v(n) = ceiling((1/3)(0+0, 1+1, 4+3, 9+6, 16+10, ...)) = (0, 1, 3, 5, 9, ...).

s[n_] := n^2; t[n_] := n (n + 1)/2; r = 1/3;

u[n_] := Floor[r*(s[n] + t[n])];

v[n_] := Ceiling[r*(s[n] + t[n])];

Table[u[n], {n, 0, 60}] (* A389142 *)

Table[v[n], {n, 0, 60}] (* A389143 *)

Cf. A000217, A000290, A111097, A387355, A389142.

Sequence in context: A118002 A069533 A054066 * A081946 A166709 A310040

Adjacent sequences: A389140 A389141 A389142 * A389144 A389145 A389146

nonn,easy

Clark Kimberling, Sep 29 2025

approved