A082108 a(n) = 4*n^2 + 6*n + 1.

1, 11, 29, 55, 89, 131, 181, 239, 305, 379, 461, 551, 649, 755, 869, 991, 1121, 1259, 1405, 1559, 1721, 1891, 2069, 2255, 2449, 2651, 2861, 3079, 3305, 3539, 3781, 4031, 4289, 4555, 4829, 5111, 5401, 5699, 6005, 6319, 6641, 6971, 7309, 7655, 8009, 8371

Offset: 0

Author

Paul Barry, Apr 03 2003

Keywords

• easy
• nonn

Comments

a(n) is the sum of the numerator and denominator of (n+1)/(2*n) + (n+2)/(2*(n+1)); all fractions are reduced and n > 0. - J. M. Bergot, Jun 14 2017

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A028387, A069131.

Programs

Magma

[4*n^2+6*n+1: n in [0..60]]; // G. C. Greubel, Dec 22 2022

Mathematica

(* Programs from Michael De Vlieger, Jun 15 2017 *)
Table[4n^2 +6n +1, {n,0,50}]
LinearRecurrence[{3,-3,1}, {1,11,29}, 51]
CoefficientList[Series[(1+8*x-x^2)/(1-x)^3, {x,0,50}], x] (* End *)

PARI

a(n)=4*n^2+6*n+1 \\ Charles R Greathouse IV, Oct 07 2015

SageMath

[4*n^2+6*n+1 for n in range(61)] # G. C. Greubel, Dec 22 2022

Formula

a(n) = a(n-1) + 8*n + 2. - Vincenzo Librandi, Aug 08 2010

From Michael De Vlieger, Jun 15 2017: (Start)

G.f.: (1 + 8*x - x^2)/(1 - x)^3.

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

a(n) = A016742(n) + A008588(n) + 1. - Felix Fröhlich, Jun 16 2017

Sum_{k=1..n} a(k-1)/(2*k)! = 1 - 1/(2*n)!. - Robert Israel, Jul 19 2017

E.g.f.: (1 + 10*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022

Extensions

Incorrect formula and useless examples deleted by R. J. Mathar, Aug 31 2010