A392105 Upper (1,1/2) midsequence of (n^2) and (n^3); see Comments.

0, 2, 8, 23, 48, 88, 144, 221, 320, 446, 600, 787, 1008, 1268, 1568, 1913, 2304, 2746, 3240, 3791, 4400, 5072, 5808, 6613, 7488, 8438, 9464, 10571, 11760, 13036, 14400, 15857, 17408, 19058, 20808, 22663, 24624, 26696, 28880, 31181, 33600, 36142, 38808, 41603

Offset

0,2

Comments

Suppose that s = (s(n)) and t = (t(n)) are sequences of numbers and h > 0 and k > 0. he 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). The lower (1,1/2) midsequence of (n^2) and (n^3) is |A269429|.

Links

Table of n, a(n) for n=0..43.

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5), with (a(0),...,a(4)) = (0,2,8,23,48).

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

Example

s = A000290 = (0, 1, 4, 9, 16, 25, 36, ...).
t = A000578 = (0, 1, 8, 27, 64, 125, 216, ...).
u(n) = (0, 1, 8, 22, 48, 87, 144, 220, 320, 445, 600, 786, 1008, ...).
v(n) = (0, 2, 8, 23, 48, 88, 144, 221, 320, 446, 600, 787, 1008, ...).

Mathematica

f[n_] := n^2; g[n_] := n^3; r = 1; s = 1/2;
u[n_] := Floor[r*f[n] + s*g[n]];
v[n_] := Ceiling[r*f[n] + s*g[n]];
Table[u[n], {n, 0, z}]   (* |A269429|*)
Table[v[n], {n, 0, z}]   (* A392105 *)
(* Also *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 8, 22, 48}, 30]   (* |A269429| *)
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 2, 8, 23, 48}, 30]   (* A392105 *)

Prog

(Python)
def A392105(n): return sum(divmod(n**2*(n+2), 2)) # Chai Wah Wu, Feb 23 2026

Crossrefs

Cf. A000290, A000578, A269429.

Sequence in context: A178129 A203298 A161463 * A190021 A014285 A331756

Adjacent sequences: A392102 A392103 A392104 * A392106 A392107 A392108

Keyword

nonn,easy

Author

Clark Kimberling, Feb 12 2026

Status

approved