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A085027 a(n) = (4*n+3)*(4*n+7).

%I A085027 #32 Oct 08 2023 04:45:40
%S A085027 21,77,165,285,437,621,837,1085,1365,1677,2021,2397,2805,3245,3717,
%T A085027 4221,4757,5325,5925,6557,7221,7917,8645,9405,10197,11021,11877,12765,
%U A085027 13685,14637,15621,16637,17685,18765,19877,21021,22197,23405,24645,25917,27221,28557,29925,31325,32757,34221
%N A085027 a(n) = (4*n+3)*(4*n+7).
%C A085027 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.
%C A085027 With A003185(n) = (4*n+1)*(4*n+5), a bisection of A078371(n) which is a bisection of A061037(n+2).
%C A085027 A quadrisection of A061037(n+2). After A002378(n), A003185(n) and A000466(n+1). - _Paul Curtz_, Mar 30 2011
%H A085027 Vincenzo Librandi, <a href="/A085027/b085027.txt">Table of n, a(n) for n = 0..10000</a>
%H A085027 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A085027 a(n) = 16*n^2+40*n+21. - _Vincenzo Librandi_, Aug 13 2011
%F A085027 From _Colin Barker_, Jul 11 2012: (Start)
%F A085027 a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).
%F A085027 G.f.: (21+14*x-3*x^2)/(1-x)^3. (End)
%F A085027 E.g.f.: (21 +56*x +16*x^2)*exp(x). - _G. C. Greubel_, Sep 20 2018
%F A085027 From _Amiram Eldar_, Oct 08 2023: (Start)
%F A085027 Sum_{n>=0} 1/a(n) = 1/12.
%F A085027 Sum_{n>=0} (-1)^n/a(n) = Pi/(8*sqrt(2)) + log(sqrt(2)-1)/(4*sqrt(2)) - 1/12. (End)
%e A085027 21 = (3)(7), 77 = (7)(11), 165 = (11)(15), 285 = (15)(19), 437 = (19)(23)...
%t A085027 Table[(4*n + 3) (4*n + 7), {n, 0, 45}]
%o A085027 (Magma) [16*n^2+40*n+21: n in [0..35]]; // _Vincenzo Librandi_, Aug 13 2011
%o A085027 (PARI) a(n)=(4*n+3)*(4*n+7) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A085027 Cf. A000466, A002378, A003185, A061037, A078371.
%K A085027 nonn,easy
%O A085027 0,1
%A A085027 _Gary W. Adamson_, Jun 19 2003