A114949 a(n) = n^2 + 6.

6, 7, 10, 15, 22, 31, 42, 55, 70, 87, 106, 127, 150, 175, 202, 231, 262, 295, 330, 367, 406, 447, 490, 535, 582, 631, 682, 735, 790, 847, 906, 967, 1030, 1095, 1162, 1231, 1302, 1375, 1450, 1527, 1606, 1687, 1770, 1855, 1942, 2031, 2122, 2215, 2310, 2407, 2506

Offset: 0

Author

Cino Hilliard, Feb 21 2006

Keywords

• nonn
• easy

Comments

2/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 = 2r/3, r2 = r - r1 = r/3. See the MathWorld link for such a Pappus chain. The clockwise chain companion has circle radii R'(n)/r = 2/A222465(n), n >= 0. - Wolfdieter Lang, Mar 01 2013

Examples

			The arbelos chain defined in a comment above has circle radii [1/3, 2/7, 1/5, 2/15, 1/11, 2/31, 1/21, 2/55, 1/35, 2/87, 1/53,...], for n >= 0. - _Wolfdieter Lang_, Mar 01 2013
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Pappus chain.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A002522, A016742, A059100, A082044, A087475, A114964 (see comment), A117619, A117950, A117951, A222465.

Programs

Maple

A114949:=n->n^2+6: seq(A114949(n), n=0..100); # Wesley Ivan Hurt, Apr 28 2017

Mathematica

Range[0, 49]^2 + 6 (* Alonso del Arte, Jan 30 2013 *)

Formula

From R. J. Mathar, May 17 2009: (Start)

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

G.f.: -(6 - 11*x + 7*x^2)/(x - 1)^3. (End)

a(n) = 2*n + a(n - 1) - 1, with n > 0, a(0)=6. - Vincenzo Librandi, Nov 13 2010

a(n) = A000290(n) + 6. - Omar E. Pol, Mar 02 2013

a(n) = ((n-2)^3 + (n-1)^3 + n^3 + (n+1)^3 + (n+2)^3)/(5*n) for n>=1. - Bruno Berselli, May 12 2014

For n >= 1, a(n) = (A016742(n) + A082044(n) - 1)/A000290(n). - Bruce J. Nicholson, Apr 19 2017

From Amiram Eldar, Nov 02 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/12.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/12. (End)

From Amiram Eldar, Feb 05 2024: (Start)

Product_{n>=0} (1 - 1/a(n)) = sqrt(5/6)*sinh(sqrt(5)*Pi)/sinh(sqrt(6)*Pi).

Product_{n>=0} (1 + 1/a(n)) = sqrt(7/6)*sinh(sqrt(7)*Pi)/sinh(sqrt(6)*Pi). (End)

E.g.f.: exp(x)*(6 + x + x^2). - Elmo R. Oliveira, Jan 17 2025