|
%I #32 Jan 07 2021 05:05:11
%S 45,221,525,957,1517,2205,3021,3965,5037,6237,7565,9021,10605,12317,
%T 14157,16125,18221,20445,22797,25277,27885,30621,33485,36477,39597,
%U 42845,46221,49725,53357,57117,61005,65021,69165,73437,77837,82365
%N a(n) = (8*n+5)*(8*n+9).
%C From _Miklos Kristof_, Nov 03 2008: (Start)
%C f(y) = y^4*(1 + y^4) = y^4 - y^8 + y^12 - y^16 + y^20 - y^24 + ...
%C Integral_{y} f(y) dy = y^5/5 - y^9/9 + y^13/13 - y^17/17 + y^21/21 - y^25/25 + ...
%C Integral_{y=0..1} f(y) dy = 1/5 - 1/9 + 1/13 - 1/17 + 1/21 - 1/25 + ...
%C = (9 - 5)/(5*9) + (17 - 13)/(13*17) + (25 - 21)/(21*25) + ...
%C = 4/(5*9) + 4/(13*17) + 4/(21*25) + ...
%C Integral_{y=0..1} f(y) dy = Sum_{m>=0} 4/((8*m+5)*(8*m+9))
%C = -(1/8)*sqrt(2)*Pi + 1 - (1/4)*sqrt(2)*log(1+sqrt(2))
%C = 0.13302701266008896241... (End)
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f: (45 + 86*x - 3*x^2)/(1-x)^3.
%F E.g.f.: (45 + 176*x + 64*x^2)*exp(x).
%F a(n) = A004770(n) * A004768(n). - _Reinhard Zumkeller_, Oct 30 2008
%p seq((8*m+5)*(8*m+9),m=0..40); # _Miklos Kristof_, Nov 03 2008
%t Table[(8n+5)(8n+9),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{45,221,525},40] (* _Harvey P. Dale_, Oct 10 2015 *)
%o (PARI) a(n)=(8*n+5)*(8*n+9) \\ _Charles R Greathouse IV_, Jun 17 2017
%K nonn,easy
%O 0,1
%A _Miklos Kristof_, Oct 29 2008
|
