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URL: https://oeis.org/A085027

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A085027
a(n) = (4*n+3)*(4*n+7).
4
21, 77, 165, 285, 437, 621, 837, 1085, 1365, 1677, 2021, 2397, 2805, 3245, 3717, 4221, 4757, 5325, 5925, 6557, 7221, 7917, 8645, 9405, 10197, 11021, 11877, 12765, 13685, 14637, 15621, 16637, 17685, 18765, 19877, 21021, 22197, 23405, 24645, 25917, 27221, 28557, 29925, 31325, 32757, 34221
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.
With A003185(n) = (4*n+1)*(4*n+5), a bisection of A078371(n) which is a bisection of A061037(n+2).
A quadrisection of A061037(n+2). After A002378(n), A003185(n) and A000466(n+1). - Paul Curtz, Mar 30 2011
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