A389320 - OEIS

%I #13 Nov 08 2025 20:41:17

%S 0,0,1,3,5,8,12,16,21,27,34,42,50,59,69,79,90,102,115,129,143,158,174,

%T 190,207,225,244,264,284,305,327,349,372,396,421,447,473,500,528,556,

%U 585,615,646,678,710,743,777,811,846,882,919,957,995,1034,1074,1114

%N Upper (3/4)-midsequence of square numbers and negative triangular numbers; see Comments.

%C 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.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-4,4,-4,4,-3,1).

%F a(n) = ceiling((3/8)*(n^2 - n)).

%F a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 3*a(n-8) + a(n-9), with (a(0),...,a(8)) = (0, 0, 1, 3, 5, 8, 12, 16, 21).

%e s = (n^2) = A000290 = (0, 1, 4, 9, 16, 25, ...).

%e t = -(n*(n+1)/2) = -A000217 = (0, -1, -3, -6, -10, -15, ...).

%e u(n) = floor((3/4)*(0-0, 1-1, 4-3, 9-6, 16-10, ...)) = (0, 0, 0, 2, 4, 7, 11, 15, ...).

%e v(n) = ceiling((3/4)*(0-0, 1-1, 4-3, 9-6, 16-10, ...)) = (0, 0, 1, 3, 5, 8, 12, 16, ...).

%t s[n_] := n^2; t[n_] := - n (n + 1)/2; r = 3/4;

%t u[n_] := Floor[r*(s[n] + t[n])];

%t v[n_] := Ceiling[r*(s[n] + t[n])];

%t Table[u[n], {n, 0, 60}]  (* A389319 *)

%t Table[v[n], {n, 0, 60}]  (* A389320 *)

%o (Python)

%o def A389320(n): return (3*n*(n-1)-1>>3)+1 # _Chai Wah Wu_, Nov 08 2025

%Y Cf. A000217, A000290, A387355, A389318, A389319.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Nov 02 2025