%I #32 Sep 04 2024 19:05:22

%S 4,10,40,130,340,754,1480,2650,4420,6970,10504,15250,21460,29410,

%T 39400,51754,66820,84970,106600,132130,162004,196690,236680,282490,

%U 334660,393754,460360,535090,618580,711490,814504,928330,1053700,1191370

%N a(n) = n^4 + 5*n^2 + 4.

%H Shawn A. Broyles, <a href="/A156798/b156798.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) = A002522(n)*A087475(n) = A000290(n) + A000290(A059100(n)) = A028552(A002522(n)).

%F a(n) = (n^2 + 1)*(n^2 + 4) = n^2 + (n^2 + 2)^2.

%F G.f.: 2*(2 -5*x +15*x^2 -5*x^3 +5*x^4)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009; corrected by _R. J. Mathar_, Sep 16 2009

%F a(0)=4, a(1)=10, a(2)=40, a(3)=130, a(4)=340, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, May 04 2011

%F From _Amiram Eldar_, Jan 18 2021: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + Pi*coth(Pi))/8 - Pi*tanh(Pi)/24.

%F Sum_{n>=0} (-1)^n/a(n) = 1/8 + Pi*csch(Pi)/6 - Pi*csch(Pi)*sech(Pi)/24. (End)

%F E.g.f.: (4 + 6*x + 12*x^2 + 6*x^3 + x^4)*exp(x). - _G. C. Greubel_, Jun 10 2021

%t Table[n^4+5n^2+4, {n,0,40}]

%o (Magma) [n^4+5*n^2+4: n in [0..50]];

%o (PARI) a(n)=n^4+5*n^2+4

%o (Sage) [(n^2 +1)*(n^2 +4) for n in (0..50)] # _G. C. Greubel_, Jun 10 2021

%Y Cf. A000290, A002522, A059100, A087475.

%K nonn,easy

%O 0,1

%A _Reinhard Zumkeller_, Feb 16 2009