A156798 a(n) = n^4 + 5*n^2 + 4.

4, 10, 40, 130, 340, 754, 1480, 2650, 4420, 6970, 10504, 15250, 21460, 29410, 39400, 51754, 66820, 84970, 106600, 132130, 162004, 196690, 236680, 282490, 334660, 393754, 460360, 535090, 618580, 711490, 814504, 928330, 1053700, 1191370

Offset: 0

Author

Reinhard Zumkeller, Feb 16 2009

Keywords

• nonn
• easy

Links

• Shawn A. Broyles, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000290, A002522, A059100, A087475.

Programs

Magma

[n^4+5*n^2+4: n in [0..50]];

Mathematica

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

PARI

a(n)=n^4+5*n^2+4

Sage

[(n^2 +1)*(n^2 +4) for n in (0..50)] # G. C. Greubel, Jun 10 2021

Formula

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

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

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

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

From Amiram Eldar, Jan 18 2021: (Start)

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

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

E.g.f.: (4 + 6*x + 12*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Jun 10 2021