A396812 Upper (1,1/2) midsequence of (floor[n/2]^2) and (ceiling[n/2]^2); see Comments.

0, 1, 2, 3, 6, 9, 14, 17, 24, 29, 38, 43, 54, 61, 74, 81, 96, 105, 122, 131, 150, 161, 182, 193, 216, 229, 254, 267, 294, 309, 338, 353, 384, 401, 434, 451, 486, 505, 542, 561, 600, 621, 662, 683, 726, 749, 794, 817, 864, 889, 938, 963, 1014, 1041, 1094, 1121, 1176, 1205, 1262, 1291, 1350

Offset

0,3

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,-1,1,-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, 0, 1, 3, 6, 9, 14, 17).

G.f.: x*(1 + x + 2*x^3 + x^4 + x^5)/((1 - x)^3*(1 + x)^2*(1 + x^2)).

a(n) - a(n-1) = A215495(n-1). - R. J. Mathar, Jun 30 2026

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, 6, 8, 13, 17, 24, 28, 37, 43, 54, 60, ...).
(v(n)) = (0, 1, 2, 3, 6, 9, 14, 17, 24, 29, 38, 43, 54, 61, ...).

Mathematica

z = 60; f[n_] := Floor[n/2]^2; g[n_] := Ceiling[n/2]^2;
r = 1; s = 1/2;
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}]
(* Also *)
LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {0, 0, 1, 3, 6, 8, 13}, 30]
LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {0, 1, 2, 3, 6, 9, 14, 17}, 30]

Crossrefs

Cf. A389123, A396811, A396813, A396814.

Sequence in context: A348865 A244739 A075123 * A023559 A191184 A299100

Adjacent sequences: A396808 A396809 A396811 * A396813 A396814 A396815

Keyword

nonn,easy

Author

Clark Kimberling, Jun 15 2026

Status

approved