A394246 Upper (1/3,2/3) midsequence of (floor[n/2]^2) and (ceiling[n/2]^2); see Comments.

0, 1, 1, 3, 4, 8, 9, 14, 16, 22, 25, 33, 36, 45, 49, 59, 64, 76, 81, 94, 100, 114, 121, 137, 144, 161, 169, 187, 196, 216, 225, 246, 256, 278, 289, 313, 324, 349, 361, 387, 400, 428, 441, 470, 484, 514, 529, 561, 576, 609, 625, 659, 676, 712, 729, 766, 784, 822, 841, 881, 900

Offset

0,4

Comments

Suppose that s = (s(n)) and t = (t(n)) are sequences of numbers and h > 0 and k > 0. The lower (h, k)-midsequence of s and t is floor(h*s + k*t); the upper (h, k)-midsequence of s and t is ceiling(h*s + k*t).

Links

Table of n, a(n) for n=0..60.

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

Formula

a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9), with (a(0),...,a(8)) = (0, 1, 1, 3, 4, 8, 9, 14, 16).

G.f.: (x (-1 - x^2 - x^3 - 2 x^4 - x^7))/((-1 + x)^3 (1 + x)^2 (1 + x^2 + x^4)).

Example

s(n) = A008794(n+1):  (0, 1, 1, 4, 4, 9, 9, 16, 16, ...).
t(n) = A008794(n+2):  (1, 1, 4, 4, 9, 9, 16, 16, 25, ...).
(u(n)) = (0, 0, 1, 3, 4, 7, 9, 13, 16, 22, 25, 32, 36, 44, ...).
(v(n)) = (0, 1, 1, 3, 4, 8, 9, 14, 16, 22, 25, 33, 36, 45, ...).

Mathematica

z = 60; f[n_] := Floor[n/2]^2; g[n_] := Ceiling[n/2]^2;
r = 1/3; s = 2/3;
u[n_] := Floor[r*f[n] + s*g[n]]
v[n_] := Ceiling[r*f[n] + s*g[n]]
Table[u[n], {n, 0, z}]
Table[v[n], {n, 0, z}]
(* Alternative: *)
LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 0, 1, 3, 4, 7, 9, 13, 16}, 30]
(* Alternative: *)
LinearRecurrence[{1, 1, -1, 0, 0, 1, -1, -1, 1}, {0, 1, 1, 3, 4, 8, 9, 14, 16}, 30]

Crossrefs

Cf. A389123, A396811, A146994.

Sequence in context: A047204 A050035 A306901 * A355477 A341578 A368641

Adjacent sequences: A394243 A394244 A394245 * A394247 A394248 A394249

Keyword

nonn,easy

Author

Clark Kimberling, Jun 16 2026

Status

approved