A396811 Lower (1,1/2) midsequence of (floor[n/2]^2) and (ceiling[n/2]^2); see Comments.

0, 0, 1, 3, 6, 8, 13, 17, 24, 28, 37, 43, 54, 60, 73, 81, 96, 104, 121, 131, 150, 160, 181, 193, 216, 228, 253, 267, 294, 308, 337, 353, 384, 400, 433, 451, 486, 504, 541, 561, 600, 620, 661, 683, 726, 748, 793, 817, 864, 888, 937, 963, 1014, 1040, 1093

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..54.

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, 8, 13, 17).

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

a(n) - a(n-1) = A212831(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, A396812, A396813, A396814.

Sequence in context: A337484 A393992 A139001 * A090961 A073355 A067440

Adjacent sequences: A396807 A396808 A396809 * A396812 A396813 A396814

Keyword

nonn,easy

Author

Clark Kimberling, Jun 15 2026

Status

approved