A241889 a(n) = n^2 + 23.

23, 24, 27, 32, 39, 48, 59, 72, 87, 104, 123, 144, 167, 192, 219, 248, 279, 312, 347, 384, 423, 464, 507, 552, 599, 648, 699, 752, 807, 864, 923, 984, 1047, 1112, 1179, 1248, 1319, 1392, 1467, 1544, 1623, 1704, 1787, 1872, 1959, 2048, 2139, 2232, 2327, 2424, 2523

Offset: 0

Author

Vincenzo Librandi, May 01 2014

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. similar sequences listed in A114962.

Programs

Magma

[n^2+23: n in [0..60]];

Mathematica

CoefficientList[Series[(23 - 45 x + 24 x^2)/(1 - x)^3,{x, 0, 60}], x]
Range[0, 50]^2 + 23 (* or *) LinearRecurrence[{3, -3, 1}, {23, 24, 27}, 50] (* Harvey P. Dale, May 27 2014 *)

PARI

a(n)=n^2+23 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (23 - 45*x + 24*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.

From Amiram Eldar, Nov 04 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(23)*Pi*coth(sqrt(23)*Pi))/46.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(23)*Pi*cosech(sqrt(23)*Pi))/46. (End)

E.g.f.: exp(x)*(23 + x + x^2). - Elmo R. Oliveira, Nov 29 2024