A027688 a(n) = n^2 + n + 3.

3, 5, 9, 15, 23, 33, 45, 59, 75, 93, 113, 135, 159, 185, 213, 243, 275, 309, 345, 383, 423, 465, 509, 555, 603, 653, 705, 759, 815, 873, 933, 995, 1059, 1125, 1193, 1263, 1335, 1409, 1485, 1563, 1643, 1725, 1809, 1895, 1983, 2073, 2165, 2259, 2355, 2453, 2553, 2655

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A002522, A176271.

Programs

Maple

with (combinat):seq(fibonacci(3, n)+n+2, n=0..47); # Zerinvary Lajos, Jun 07 2008

Mathematica

Table[n^2 + n + 3, {n, 0, 50}] (* Bruno Berselli, Sep 03 2018 *)

PARI

Vec((3*x^2-4*x+3)/(1-x)^3 + O(x^100)) \\ Colin Barker, Dec 29 2014

Formula

For n > 0: a(n) = A176271(n+1,2). - Reinhard Zumkeller, Apr 13 2010

a(n) = 2*n + a(n-1) (with a(0)=3). - Vincenzo Librandi, Aug 05 2010

From Colin Barker, Dec 29 2014: (Start)

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

G.f.: (3*x^2 - 4*x + 3)/(1 - x)^3. (End)

Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(11)/2)/sqrt(11). - Amiram Eldar, Jan 17 2021

E.g.f.: exp(x)*(3 + 2*x + x^2). - Elmo R. Oliveira, Oct 31 2024

Extensions

Definition and offset corrected by Franklin T. Adams-Watters, Jul 06 2009