A396811 - OEIS

%I #10 Jun 30 2026 06:17:44

%S 0,0,1,3,6,8,13,17,24,28,37,43,54,60,73,81,96,104,121,131,150,160,181,

%T 193,216,228,253,267,294,308,337,353,384,400,433,451,486,504,541,561,

%U 600,620,661,683,726,748,793,817,864,888,937,963,1014,1040,1093

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

%C 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).

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,-1,1).

%F 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).

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

%F a(n) - a(n-1) = A212831(n-1). - _R. J. Mathar_, Jun 30 2026

%e s(n) = A008794(n+1):  (0, 1, 1, 4, 4, 9, 9, 16, 16, ...).

%e t(n) = A008794(n+2):  (1, 1, 4, 4, 9, 9, 16, 16, 25, ...).

%e (u(n)) = (0, 0, 1, 3, 6, 8, 13, 17, 24, 28, 37, 43, 54, 60, ...).

%e (v(n)) = (0, 1, 2, 3, 6, 9, 14, 17, 24, 29, 38, 43, 54, 61, ...).

%t z = 60; f[n_] := Floor[n/2]^2; g[n_] := Ceiling[n/2]^2;

%t r = 1; s = 1/2;

%t u[n_] := Floor[r*f[n] + s*g[n]]

%t v[n_] := Ceiling[r*f[n] + s*g[n]]

%t Table[u[n], {n, 0, z}]

%t Table[v[n], {n, 0, z}]

%t (* Also *)

%t LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {0, 0, 1, 3, 6, 8, 13}, 30]

%t LinearRecurrence[{1, 1, -1, 1, -1, -1, 1}, {0, 1, 2, 3, 6, 9, 14, 17}, 30]

%Y Cf. A389123, A396812, A396813, A396814.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Jun 15 2026