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a(n) = (8*n+1)*(8*n+3)*(8*n+5)*(8*n+7).
1

%I #37 Sep 08 2022 08:44:29

%S 105,19305,156009,606825,1666665,3728745,7284585,12924009,21335145,

%T 33304425,49716585,71554665,99900009,135932265,180929385,236267625,

%U 303421545,383964009,479566185,591997545,723125865,874917225,1049436009,1248844905,1475404905

%N a(n) = (8*n+1)*(8*n+3)*(8*n+5)*(8*n+7).

%C From _Jaume Oliver Lafont_, May 30 2010: (Start)

%C 3 divides a(n).

%C Sum_{k>=0} 1/a(k) = Pi/(96*(2+sqrt(2))) = 0.0095849081719... [Jolley eq. 243] (End)

%D Jolley, Summation of Series, Dover (1961).

%H Vincenzo Librandi, <a href="/A001546/b001546.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F G.f.: -3*(35+6260*x+20178*x^2+6260*x^3+35*x^4)/(x-1)^5. [Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009]

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - _Wesley Ivan Hurt_, Jan 02 2017

%p A001546:=n->(8*n+1)*(8*n+3)*(8*n+5)*(8*n+7): seq(A001546(n), n=0..50); # _Wesley Ivan Hurt_, Jan 02 2017

%t Table[Times@@(8n + {1, 3, 5, 7}), {n, 0, 30}] (* _Harvey P. Dale_, Jan 09 2011 *)

%o (PARI) a(n)=(8*n+1)*(8*n+3)*(8*n+5)*(8*n+7) \\ _Charles R Greathouse IV_, Oct 03 2011

%o (Magma) [(8*n+1)*(8*n+3)*(8*n+5)*(8*n+7): n in [0..20]]; // _Vincenzo Librandi_, Oct 04 2011

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_