A393346 Lower (1,1/3) midsequence of (n^2) and (n^3); see Comments.

0, 1, 6, 18, 37, 66, 108, 163, 234, 324, 433, 564, 720, 901, 1110, 1350, 1621, 1926, 2268, 2647, 3066, 3528, 4033, 4584, 5184, 5833, 6534, 7290, 8101, 8970, 9900, 10891, 11946, 13068, 14257, 15516, 16848, 18253, 19734, 21294, 22933, 24654, 26460, 28351, 30330

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,-3,2,-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)) = (0,1,6,18,37,66). [Corrected by Georg Fischer, Feb 24 2026]

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

E.g.f.: exp(-x/2)*(3*exp(3*x/2)*(1 + x)*(x^2 + 5*x - 1) + 3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Feb 12 2026

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 A393346(n): return n**2*(n+3)//3 # Chai Wah Wu, Feb 23 2026

Crossrefs

Cf. A000290, A000578, A393347.

Sequence in context: A389615 A299272 A101853 * A132432 A005899 A261652

Adjacent sequences: A393343 A393344 A393345 * A393347 A393348 A393349

Keyword

nonn,easy

Author

Clark Kimberling, Feb 12 2026

Status

approved