A390566 Upper (1/3,1/2) midsequence of (n^2) and (n^3); see Comments.

0, 1, 6, 17, 38, 71, 120, 188, 278, 392, 534, 706, 912, 1155, 1438, 1763, 2134, 2553, 3024, 3550, 4134, 4778, 5486, 6260, 7104, 8021, 9014, 10085, 11238, 12475, 13800, 15216, 16726, 18332, 20038, 21846, 23760, 25783, 27918, 30167, 32534, 35021, 37632, 40370

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..43.

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)) = (0,1,6,17,38,71,120,188,278).

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

Mathematica

s[n_] := n^2 ; t[n_] := n^3; {u1, v1} = {1/3, 1/2};
u[n_] := Floor[u1*s[n] + v1*t[n]]
v[n_] := Ceiling[u1*s[n] + v1*t[n]]
tu = Table[u[n], {n, 0, 60}]  (* A390565 *)
tv = Table[v[n], {n, 0, 60}]  (* A390566 *)
(* Also *)
LinearRecurrence[{2, 0, -1, -1, 0, 2, -1}, {0, 0, 5, 16, 37, 70, 120}, 30]    (* A390565 *)
LinearRecurrence[{3, -3, 1, 0, 0, 1, -3, 3, -1}, {0, 1, 6, 17, 38, 71, 120, 188, 278}, 30]  (* A390566 *)

Crossrefs

Cf. A000290, A000578, A389123, A390565.

Sequence in context: A132127 A023621 A000385 * A192756 A004799 A085278

Adjacent sequences: A390563 A390564 A390565 * A390567 A390568 A390569

Keyword

nonn,easy

Author

Clark Kimberling, Nov 19 2025

Status

approved