A393347 Upper (1,1/3) midsequence of (n^2) and (n^3); see Comments.

0, 2, 7, 18, 38, 67, 108, 164, 235, 324, 434, 565, 720, 902, 1111, 1350, 1622, 1927, 2268, 2648, 3067, 3528, 4034, 4585, 5184, 5834, 6535, 7290, 8102, 8971, 9900, 10892, 11947, 13068, 14258, 15517, 16848, 18254, 19735, 21294, 22934, 24655, 26460, 28352, 30331

Offset

0,2

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,-3,2,-3,3,-1).

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6), with (a(0),...,a(5)) = (0,2,7,18,38,67).

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

Example

s = A000290 = (0, 1, 4, 9, 16, 25, 36, ...).
t = A000578 = (0, 1, 8, 27, 64, 125, 216, ...).
u(n) = (0, 1, 6, 18, 37, 66, 108, 163, 234, 324, 433, 564, 720, ...).
v(n) = (0, 2, 7, 18, 38, 67, 108, 164, 235, 324, 434, 565, 720, ...).

Mathematica

f[n_] := n^2; g[n_] := n^3; r = 1; 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}]   (* A393346 *)
Table[v[n], {n, 0, z}]   (* A393347 *)
(* Also *)
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {0, 1, 6, 18, 37, 66}, 50] (* A393346 *)
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {0, 2, 7, 18, 38, 67}, 50] (* A393347 *)

Prog

(Python)
def A393347(n): return (lambda x:x[0]+bool(x[1]))(divmod(n**2*(n+3), 3)) # Chai Wah Wu, Feb 23 2026

Crossrefs

Cf. A000290, A000578, A393346.

Sequence in context: A212685 A176854 A086741 * A229183 A051743 A054111

Adjacent sequences: A393344 A393345 A393346 * A393348 A393349 A393350

Keyword

nonn,easy

Author

Clark Kimberling, Feb 12 2026

Status

approved