A389583 Upper (1/2,1/3) midsequence of (n^2) and (n^3); see Comments.

0, 1, 5, 14, 30, 55, 90, 139, 203, 284, 384, 505, 648, 817, 1013, 1238, 1494, 1783, 2106, 2467, 2867, 3308, 3792, 4321, 4896, 5521, 6197, 6926, 7710, 8551, 9450, 10411, 11435, 12524, 13680, 14905, 16200, 17569, 19013, 20534, 22134, 23815, 25578, 27427, 29363

Offset

0,3

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..44.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,1,-3,3,-1).

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-6) - 3*a(n-7) + 3*a(n-8) - a(n-9), with (a(0),...,a(8)) = (0,1,5,14,30,55,90,139,203).

G.f.: x*(1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + 3*x^6 - x^7)/((-1 + x)^4*(1 + x + x^2 + x^3 + x^4 + x^5)).

Example

s = A000290 = (0, 1, 4, 9, 16, 25, 36, ...).
t = A000578 = (0, 1, 8, 27, 64, 125, 216, ...).
u(n) = (0, 0, 4, 13, 29, 54, 90, 138, 202, 283, ...).
v(n) = (0, 1, 5, 14, 30, 55, 90, 139, 203, 284, ...).

Mathematica

f[n_] := n^2; g[n_] := n^3; r = 1/2; s = 1/3;
u[n_] := Floor[r*f[n] + s*g[n]]
v[n_] := Ceiling[r*f[n] + s*g[n]]
Table[u[n], {n, 0, z}]   (* A389582 *)
Table[v[n], {n, 0, z}]   (* A389583 *)
(* Also *)
LinearRecurrence[{3, -3, 1, 0, 0, 1, -3, 3, -1}, {0, 0, 4, 13, 29, 54, 90, 138, 202}, 30]   (* A390582 *)
LinearRecurrence[{3, -3, 1, 0, 0, 1, -3, 3, -1}, {0, 1, 5, 14, 30, 55, 90, 139, 203}, 30]   (* A390583 *)

Crossrefs

Cf. A000290, A000578, A389582.

Sequence in context: A136135 A231673 A211653 * A096893 A231677 A231681

Adjacent sequences: A389580 A389581 A389582 * A389584 A389585 A389586

Keyword

nonn,easy

Author

Clark Kimberling, Dec 07 2025

Status

approved