A389316 Lower (2/3)-midsequence of square numbers and triangular numbers; see Comments.

0, 1, 4, 10, 17, 26, 38, 51, 66, 84, 103, 124, 148, 173, 200, 230, 261, 294, 330, 367, 406, 448, 491, 536, 584, 633, 684, 738, 793, 850, 910, 971, 1034, 1100, 1167, 1236, 1308, 1381, 1456, 1534, 1613, 1694, 1778, 1863, 1950, 2040, 2131, 2224, 2320, 2417

Offset

0,3

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.

Links

Table of n, a(n) for n=0..49.

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Formula

a(n) = floor((3*n^2 + n)/3).

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).

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

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((2/3)(0+0, 1+1, 4+3, 9+6, 16+10, ...)) = (0, 1, 4, 10, 17, ...).
v(n) = ceiling((2/3)(0+0, 1+1, 4+3, 9+6, 16+10, ...)) = (0, 2, 5, 10, 18, ...).

Mathematica

s[n_] := n^2; t[n_] := n (n + 1)/2; r = 2/3;
u[n_] := Floor[r*(s[n] + t[n])];
v[n_] := Ceiling[r*(s[n] + t[n])];
Table[u[n], {n, 0, 60}]  (* A389316 *)
Table[v[n], {n, 0, 60}]  (* A389317 *)

Crossrefs

Cf. A000217, A000290, A387355, A389142, A389317.

Sequence in context: A009860 A294249 A138105 * A213398 A002442 A350143

Adjacent sequences: A389313 A389314 A389315 * A389317 A389318 A389319

Keyword

nonn,easy

Author

Clark Kimberling, Sep 29 2025

Status

approved