A242669 a(n) = n*floor(n/3).

0, 0, 0, 3, 4, 5, 12, 14, 16, 27, 30, 33, 48, 52, 56, 75, 80, 85, 108, 114, 120, 147, 154, 161, 192, 200, 208, 243, 252, 261, 300, 310, 320, 363, 374, 385, 432, 444, 456, 507, 520, 533, 588, 602, 616, 675, 690, 705, 768, 784, 800, 867, 884, 901, 972, 990

Offset: 0

Author

Bruno Berselli, Jul 01 2014

Keywords

• nonn
• easy

Comments

For n = 0, 1, 2, 4, 8, 49, 98, 676, 1352, 9409, 18818, 131044, 262088, 1825201, 3650402, ... a(n) is a square.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Crossrefs

Cf. A000290 (n^2), A010762 (floor(n/2)*floor(n/3)), A093353 (n*floor(n/2)), A213033 (n*floor(n/2)*floor(n/3)), A233035 (n*floor(n/4)).

Cf. A033428, A045944, A049451.

Cf. A002264 (floor(n/3)).

Programs

Magma

[n*Floor(n/3): n in [0..60]];

Mathematica

Table[n Floor[n/3], {n, 0, 60}]

PARI

a(n)=n\3*n \\ Charles R Greathouse IV, Oct 07 2015

Sage

[n*floor(n/3) for n in (0..60)];

Formula

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

a(3m) = A033428(m), a(3m+1) = A049451(m), a(3m+2) = A045944(m).

Sum_{n>=3} (-1)^(n+1)/a(n) = 9/4 + Pi^2/36 - Pi/(2*sqrt(3)) - 2*log(2). - Amiram Eldar, Mar 30 2023