OFFSET
0,1
COMMENTS
1 = 3/7 + Sum_{n>=1} 16/a(n) = 3/7 + 16/77 + 16/165 + 16/285...+...; with partial sums: 3/7, 7/11, 11/15, 15/19, 19/23, ...(4n+3)/(4n+7), ... ==> 1.
A quadrisection of A061037(n+2). After A002378(n), A003185(n) and A000466(n+1). - Paul Curtz, Mar 30 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 16*n^2+40*n+21. - Vincenzo Librandi, Aug 13 2011
From Colin Barker, Jul 11 2012: (Start)
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).
G.f.: (21+14*x-3*x^2)/(1-x)^3. (End)
E.g.f.: (21 +56*x +16*x^2)*exp(x). - G. C. Greubel, Sep 20 2018
From Amiram Eldar, Oct 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 1/12.
Sum_{n>=0} (-1)^n/a(n) = Pi/(8*sqrt(2)) + log(sqrt(2)-1)/(4*sqrt(2)) - 1/12. (End)
EXAMPLE
21 = (3)(7), 77 = (7)(11), 165 = (11)(15), 285 = (15)(19), 437 = (19)(23)...
MATHEMATICA
Table[(4*n + 3) (4*n + 7), {n, 0, 45}]
PROG
(Magma) [16*n^2+40*n+21: n in [0..35]]; // Vincenzo Librandi, Aug 13 2011
(PARI) a(n)=(4*n+3)*(4*n+7) \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jun 19 2003
STATUS
approved