A057781 a(n) = n^4+4 = (n^2-2*n+2)*(n^2+2*n+2) = ((n-1)^2+1)*((n+1)^2+1).

4, 5, 20, 85, 260, 629, 1300, 2405, 4100, 6565, 10004, 14645, 20740, 28565, 38420, 50629, 65540, 83525, 104980, 130325, 160004, 194485, 234260, 279845, 331780, 390629, 456980, 531445, 614660, 707285, 810004, 923525, 1048580, 1185925

Offset: 0

Author

Henry Bottomley, Nov 04 2000

Keywords

• nonn
• easy

References

• Donald E. Knuth, The Art of Computer Programming, Addison-Wesley, Reading, MA, 1997, Vol. 1, exercise 1.2.1, Nr. 11, p. 19. [From Reinhard Zumkeller, Apr 11 2010]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A016755, A033999, A053755.

Cf. A000583, A002522, A002523, A272298.

Programs

Magma

[n^4+4: n in [0..40]]; // Vincenzo Librandi, Sep 07 2011

Mathematica

Table[n^4+4,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{4,5,20,85,260},40] (* Harvey P. Dale, Aug 20 2020 *)

PARI

first(m)=vector(m,i,i--;i^4+4) \\ Anders Hellström, Sep 15 2015

Formula

G.f.: -(5*x^4-5*x^3+35*x^2-15*x+4) / (x-1)^5. - Colin Barker, Mar 29 2013

a(n) = A002523(n) + 3.

a(n) = A002522(n-1) * A002522(n+1).

Sum_{k=0..n} A033999(k)*A016755(k)/a(k) = A033999(n)*(n+1)/A053755(n+1), see Knuth reference. - Reinhard Zumkeller, Apr 11 2010

a(n) = (n^2)^2 + 2^2 = (n^2-2)^2 + (2*n)^2. - Thomas Ordowski, Sep 15 2015

a(n) = A272298(3*n)/3^4. - Bruno Berselli, Apr 29 2016

Sum_{n>=0} 1/a(n) = (Pi*coth(Pi) + 1)/8. - Amiram Eldar, Oct 04 2021