A177071 a(n) = (7*n + 3)*(7*n + 4).

12, 110, 306, 600, 992, 1482, 2070, 2756, 3540, 4422, 5402, 6480, 7656, 8930, 10302, 11772, 13340, 15006, 16770, 18632, 20592, 22650, 24806, 27060, 29412, 31862, 34410, 37056, 39800, 42642, 45582, 48620, 51756, 54990, 58322, 61752, 65280, 68906, 72630, 76452

Offset: 0

Author

Vincenzo Librandi, May 31 2010

Keywords

• nonn
• easy

Comments

Cf. Zumkeller's contribution in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2, therefore a(n) = 49*A002061(n+1) - 37. - Bruno Berselli, Aug 24 2010

Links

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

Crossrefs

Cf. A002061, A017017, A017029, A061792, A177059.

Programs

Mathematica

Table[(7n+3)(7n+4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{12,110,306},40] (* Harvey P. Dale, Oct 09 2011 *)

PARI

a(n)=2*binomial(7*n+4,2) \\ Charles R Greathouse IV, Jan 11 2012

Formula

a(n) = 98*n + a(n-1) with n > 0, a(0)=12.

From Harvey P. Dale, Oct 09 2011: (Start)

a(0)=12, a(1)=110, a(2)=306, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: -((2*(x+6)*(6*x+1))/(x-1)^3). (End)

From Amiram Eldar, Feb 19 2023: (Start)

a(n) = A017017(n)*A017029(n).

Sum_{n>=0} 1/a(n) = tan(Pi/14)*Pi/7.

Product_{n>=0} (1 - 1/a(n)) = sec(Pi/14)*cos(sqrt(5)*Pi/14).

Product_{n>=0} (1 + 1/a(n)) = sec(Pi/14)*cosh(sqrt(3)*Pi/14). (End)

From Elmo R. Oliveira, Oct 27 2024: (Start)

E.g.f.: exp(x)*(12 + 49*x*(2 + x)).

a(n) = 2*A061792(n). (End)

Extensions

Edited by N. J. A. Sloane, Jun 22 2010