A162940 a(n) = binomial(n+1,2)*6^2.

0, 36, 108, 216, 360, 540, 756, 1008, 1296, 1620, 1980, 2376, 2808, 3276, 3780, 4320, 4896, 5508, 6156, 6840, 7560, 8316, 9108, 9936, 10800, 11700, 12636, 13608, 14616, 15660, 16740, 17856, 19008, 20196, 21420, 22680, 23976, 25308, 26676, 28080, 29520, 30996

Offset: 0

Author

Zerinvary Lajos, Jul 18 2009, Jul 19 2009

Keywords

• nonn
• easy

Comments

Number of n permutations (n>=2) of 7 objects s, t, u, v, z, x, y with repetition allowed, containing n-2 u's. Example: If n=2 then n-2 = zero (0) u, a(1)=36 because we have ss, st, sv, sz, sx, sy, ts, tt, tv, tz, tx, ty, vs, vt, vv, vz, vx, vy, zs, zt, zv, zz, zx, zy, xs, xt, xv, xz, xx, xy, ys, yt, yv, yz, yx, yy. If n=3 then n-2 = one (1) u, a(2) = 108, >> ssu, stu, etc. If n=4 then n-2 = two (2) u, a(2)= 216, >> ssuu, stuu, ..., txuu, etc. If n=5 then n-2 = three (3) u, a(3)=360, >> ssuuu, stuuu, ..., txuuu, etc.

Links

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

Crossrefs

Cf. A000217, A002378, A046092, A027468, A035008, A123296.

Cf. A049598, A163758.

Programs

Mathematica

Table[Binomial[n + 1, 2]*6^2, {n, 0, 58}]

PARI

a(n)=18*n*(n+1) \\ Charles R Greathouse IV, Jun 16 2017

Formula

From Amiram Eldar, Sep 01 2022: (Start)

Sum_{n>=1} 1/a(n) = 1/18.

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/9 - 1/18. (End)

From Amiram Eldar, Feb 22 2023: (Start)

a(n) = 18*n*(n+1) = 36*A000217(n) = 18*A002378(n).

Product_{n>=1} (1 - 1/a(n)) = -(18/Pi)*cos(sqrt(11)*Pi/6).

Product_{n>=1} (1 + 1/a(n)) = (18/Pi)*cos(sqrt(7)*Pi/6). (End)

From Elmo R. Oliveira, Nov 29 2024: (Start)

G.f.: 36*x/(1-x)^3.

E.g.f.: 18*x*(2 + x)*exp(x).

a(n) = 3*A049598(n) = 2*A163758(n).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)