A090288 a(n) = 2*n^2 + 6*n + 2.

2, 10, 22, 38, 58, 82, 110, 142, 178, 218, 262, 310, 362, 418, 478, 542, 610, 682, 758, 838, 922, 1010, 1102, 1198, 1298, 1402, 1510, 1622, 1738, 1858, 1982, 2110, 2242, 2378, 2518, 2662, 2810, 2962, 3118, 3278, 3442, 3610, 3782, 3958, 4138, 4322, 4510

Offset: 0

Author

Philippe Deléham, Jan 25 2004

Keywords

• nonn
• easy

Comments

Values of polynomial K_2 related to A090285: a(n) = K_2(n) = Sum_{k>=0} A090285(2,k)*2^k*binomial(n,k).

Numbers k such that 2*k+5 is a square. - Vincenzo Librandi, Oct 10 2013

a(n) is the area of a triangle with vertices at (b(n-2),b(n-1)), (b(n),b(n+1)), and (b(n+2),B(n+3)) for b(k)=A000292(k) with n>1. - J. M. Bergot, Mar 23 2017

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. A028387, A090285.

Programs

GAP

List([0..50], n-> 2*(1+3*n+n^2)); # G. C. Greubel, May 31 2019

Magma

[2*(1+3*n+n^2): n in [0..50]]; // G. C. Greubel, May 31 2019

Mathematica

Table[2*(n^2 +3*n +1), {n, 0, 50}] (* Vincenzo Librandi, Oct 10 2013 *)
LinearRecurrence[{3,-3,1},{2,10,22},50] (* Harvey P. Dale, May 04 2017 *)

PARI

a(n)=2*n^2+6*n+2 \\ Charles R Greathouse IV, Sep 24 2015

Sage

[2*(1+3*n+n^2) for n in (0..50)] # G. C. Greubel, May 31 2019

Formula

a(n) = 2*A028387(n).

G.f.: 2*(1 +2*x -x^2)/(1-x)^3. - R. J. Mathar, Apr 02 2008

E.g.f.: 2*(1 +4*x +x^2)*exp(x). - G. C. Greubel, Jul 13 2017

Sum_{n>=0} 1/a(n) = 1/2 + Pi*tan(sqrt(5)*Pi/2)/(2*sqrt(5)). - Amiram Eldar, Dec 23 2022

Extensions

Corrected by T. D. Noe, Nov 12 2006