A390345 Upper (2/3)-midsequence of binomial(n,3) and binomial(n,2); see Comments.

0, 0, 1, 3, 7, 14, 24, 38, 56, 80, 110, 147, 191, 243, 304, 374, 454, 544, 646, 760, 887, 1027, 1181, 1350, 1534, 1734, 1950, 2184, 2436, 2707, 2997, 3307, 3638, 3990, 4364, 4760, 5180, 5624, 6093, 6587, 7107, 7654, 8228, 8830, 9460, 10120, 10810, 11531

Offset

0,4

Comments

Suppose that s = (s(n)) and t = (t(n)) are sequences of numbers and r > 0. The lower (r)-midsequence of s and t is given by u = floor(r*(s + t)); the upper r-midsequence of s and t is given by v = ceiling(r*(s + t)). If s and t are linearly recurrent and r is rational, then u and v are linearly recurrent.

Links

Table of n, a(n) for n=0..47.

Index entries for linear recurrences with constant coefficients, signature (4,-6,3,3,-6,3,3,-6,4,-1).

Formula

a(n) = ceiling((n^3 - n)/9).

a(n) = 4*a(n-1) - 6*a(n-2) + 3*a(n-3) + 3*a(n-4) - 6*a(n-5) + 3*a(n-6) + 3*a(n-7) - 6*a(n-8) + 4*a(n-9) - a(n-10), with (a(0),...,a(9)) = (0, 0, 1, 3, 7, 14, 24, 38, 56, 80).

G.f.: x^2*(1 - x + x^2 + x^3 - 2*x^4 + 2*x^5)/((1 - x)^4*(1 + x^3 + x^6)). - Stefano Spezia, Dec 27 2025

Example

s = binomial(n,3) = (0, 0, 0, 1, 4, 10, 20, 35, 56, ...).
t = binomial(n,2) = (0, 0, 1, 3, 6, 10, 15, 21, 28, ...).
u(n) = floor((2/3)*(0+0, 0+0, 0+1, 1+3, 4+6, 10+10, ...)) = (0, 0, 0, 2, 6, 13, 23, 37, ...).
v(n) = ceiling((2/3)*(0+0, 0+0, 0+1, 1+3, 4+6, 10+10, ...)) = (0, 0, 1, 3, 7, 14, 24, 38, ...).

Mathematica

s[n_] := Binomial[n, 3] ; t[n_] := Binomial[n, 2]; r = 2/3;
u[n_] := Floor[r*(s[n] + t[n])];
v[n_] := Ceiling[r*(s[n] + t[n])];
Table[u[n], {n, 0, 60}]  (* A011849 *)
Table[v[n], {n, 0, 60}]  (* A390345 *)

Prog

(Python)
def A390345(n): return (lambda d:d[0]+bool(d[1]))(divmod(n*(n**2-1), 9)) # Chai Wah Wu, Nov 11 2025

Crossrefs

Cf. A000217, A000292, A011849, A390344.

Sequence in context: A173247 A123386 A390347 * A060999 A089187 A333980

Adjacent sequences: A390342 A390343 A390344 * A390346 A390347 A390348

Keyword

nonn,easy

Author

Clark Kimberling, Nov 09 2025

Status

approved