A028878 a(n) = (n+3)^2 - 6.

3, 10, 19, 30, 43, 58, 75, 94, 115, 138, 163, 190, 219, 250, 283, 318, 355, 394, 435, 478, 523, 570, 619, 670, 723, 778, 835, 894, 955, 1018, 1083, 1150, 1219, 1290, 1363, 1438, 1515, 1594, 1675, 1758, 1843, 1930, 2019, 2110, 2203, 2298, 2395

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

a(n-3) = n^2 - 6, n>=0, with a(-3) = -6, a(-2) = -5, a(-1) = -2 gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 24 for b = 2*n. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 16 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290.

Programs

Maple

A028878:=n->(n+3)^2-6: seq(A028878(n), n=0..100); # Wesley Ivan Hurt, Apr 28 2017

Mathematica

Table[(n + 3)^2 - 6, {n,0,50}] (* G. C. Greubel, Aug 19 2017 *)

PARI

a(n)=(n+3)^2-6 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = a(n-1) + 2*n + 5 (with a(0)=3). - Vincenzo Librandi, Aug 05 2010

From Bruno Berselli, Sep 02 2011: (Start)

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

a(n) = a(-n-6).

a(n) mod (n+1) = n-1. (End)

a(n) = A000290(n+3) - 6. - Omar E. Pol, Dec 12 2012

E.g.f.: (x^2 + 7*x + 3)*exp(x). - G. C. Greubel, Aug 19 2017

From Amiram Eldar, Nov 04 2020: (Start)

Sum_{n>=0} 1/a(n) = (47 - 5*sqrt(6)*Pi*cot(sqrt(6)*Pi))/60.

Sum_{n>=0} (-1)^n/a(n) = (-23 + 5*sqrt(6)*Pi*cosec(sqrt(6)*Pi))/60. (End)

From Amiram Eldar, Feb 05 2024: (Start)

Product_{n>=0} (1 - 1/a(n)) = (5*sqrt(2/21)/3)*sin(sqrt(7)*Pi)/sin(sqrt(6)*Pi).

Product_{n>=0} (1 + 1/a(n)) = sqrt(15/2)*sin(sqrt(5)*Pi)/sin(sqrt(6)*Pi). (End)

Extensions

Definition corrected by Omar E. Pol, Jul 27 2009