A389318 Upper (1/4)-midsequence of square numbers and (-1)*(triangular numbers); see Comments.

0, 0, 1, 1, 2, 3, 4, 6, 7, 9, 12, 14, 17, 20, 23, 27, 30, 34, 39, 43, 48, 53, 58, 64, 69, 75, 82, 88, 95, 102, 109, 117, 124, 132, 141, 149, 158, 167, 176, 186, 195, 205, 216, 226, 237, 248, 259, 271, 282, 294, 307, 319, 332, 345, 358, 372, 385, 399, 414

Offset

0,5

Comments

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. Here s = A000290 (squares) and t = (-1)*A000217 (negative triangular numbers). Other (r)-midsequences for the same s and t are as follows:

lower (1/2)-midsequence: A011848

upper (1/2)-midsequence: A054925

lower (3/2)-midsequence: A281026 except for initial 0

lower (5/2)-midsequence: A332495 except for initial 0

lower (1/3)-midsequence: A130518

upper (1/3)-midsequence: A007997

lower (2/3)-midsequence: A128422

upper (2/3)-midsequence: A007980

lower (1/4)-midsequence: A011861

lower (3/4)-midsequence: A389319

upper (3/4)-midsequence: A389320

Links

Table of n, a(n) for n=0..58.

Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-4,4,-4,4,-3,1).

Formula

a(n) = ceiling((n^2 - n)/8).

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).

G.f.: -x^2*(1 - 2*x + 3*x^2 - 3*x^3 + 3*x^4 - 2*x^5 + x^6)/((-1 + x)^3*(1 + x^2 + x^4 + x^6)).

Example

s = (n^2) = A000290 = (0, 1, 4, 9, 16, 25, ...).
t = (n*(n+1)/2) = A000217 = (0, 1, 3, 6, 10, 15, ...).
u(n) = floor((1/4)(0-0, 1-1, 4-3, 9-6, 16-10, ...)) = (0, 0, 0, 0, 1, 2, 3, 5, ...).
v(n) = ceiling((1/4)(0-0, 1-1, 4-3, 9-6, 16-10, ...)) = (0, 0, 1, 1, 2, 3, 4, 6, ...).

Mathematica

s[n_] := n^2; t[n_] := -n (n + 1)/2; r = 1/4;
u[n_] := Floor[r*(s[n] + t[n])]
v[n_] := Ceiling[r*(s[n] + t[n])]
Table[u[n], {n, 0, 60}]
Table[v[n], {n, 0, 60}]

Prog

(Python)
def A389318(n): return (n*(n-1)-1>>3)+1 # Chai Wah Wu, Oct 07 2025

Crossrefs

Cf. A000217, A000290, A387355, A389142, A389316.

Sequence in context: A162293 A233459 A145803 * A018629 A018357 A061489

Adjacent sequences: A389315 A389316 A389317 * A389319 A389320 A389321

Keyword

nonn,easy

Author

Clark Kimberling, Oct 03 2025

Status

approved