A241748 a(n) = n^2 + 12.

12, 13, 16, 21, 28, 37, 48, 61, 76, 93, 112, 133, 156, 181, 208, 237, 268, 301, 336, 373, 412, 453, 496, 541, 588, 637, 688, 741, 796, 853, 912, 973, 1036, 1101, 1168, 1237, 1308, 1381, 1456, 1533, 1612, 1693, 1776, 1861, 1948, 2037, 2128, 2221, 2316, 2413, 2512

Offset: 0

Author

Vincenzo Librandi, Apr 30 2014

Keywords

• nonn
• easy

Comments

3/a(n) = R(n)/r, n >= 0, with R(n) the n-th radius of the counterclockwise Pappus chain of the arbelos with semicircle radii r, r1 = 3*r/4, r2 = r - r1 = r/4. See a comment on A114949 also for the MathWorld Pappus chain link. - Wolfdieter Lang, Jun 29 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Kival Ngaokrajang, Illustration of Pappus chain (3/4, 1/4).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. similar sequences listed in A114962.

Cf. A114964 (see comment), A114949.

Programs

Magma

[n^2+12: n in [0..60]];

Mathematica

Table[n^2 + 12, {n, 0, 60}]

PARI

a(n)=n^2+12 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (12-23*x+13*x^2)/(1-x)^3.

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

(2*n)*a(n) = (n+2)^3 + (n-2)^3; also, 2*a(n) = (n+sqrt(12))^2 + (n-sqrt(12))^2. - Bruno Berselli, Mar 13 2015

From Amiram Eldar, Nov 02 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(12)*Pi*coth(sqrt(12)*Pi))/24.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(12)*Pi*cosech(sqrt(12)*Pi))/24. (End)

E.g.f.: exp(x)*(12 + x + x^2). - Elmo R. Oliveira, Nov 29 2024