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 (3,-4,4,-3,1).

a(n) = ceiling((3*n^2 + n)/4).

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

G.f.: -x*(1 + x + x^3)/((-1 + x)^3*(1 + 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((1/2)*(0+0, 1+1, 4+3, 9+6, 16+10, ...)) = (0, 1, 3, 7, 13, ...).

v(n) = ceiling((1/2)*(0+0, 1+1, 4+3, 9+6, 16+10, ...)) = (0, 1, 4, 8, 13, ...).

s[n_] := n^2; t[n_] := n (n + 1)/2; 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}] (* A330707 *)

Table[v[n], {n, 0, 60}] (* A387355 *)

(Python)

def A387355(n): return (n*(3*n+1)-1>>2)+1 # Chai Wah Wu, Nov 08 2025

Cf. A000217, A000290, A330707.

Sequence in context: A170907 A143978 A362016 * A071994 A023661 A157130

Adjacent sequences: A387352 A387353 A387354 * A387356 A387357 A387358

nonn,easy

Clark Kimberling, Sep 24 2025

approved