A001589 a(n) = 4^n + n^4.

1, 5, 32, 145, 512, 1649, 5392, 18785, 69632, 268705, 1058576, 4208945, 16797952, 67137425, 268473872, 1073792449, 4295032832, 17179952705, 68719581712, 274878037265, 1099511787776, 4398046705585, 17592186278672

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is prime if and only if n = 1. - Reinhard Zumkeller, May 24 2009

The statement above (and the corollary that 5 is the only prime term in this sequence) can be proved using Sophie Germain's identity x^4 + 4y^4 = (x^2 + 2xy + 2y^2)(x^2 - 2xy + 2y^2). - Alonso del Arte, Oct 31 2013 [exponents corrected by Thomas Ordowski, Nov 29 2017]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Notes, Mathematical Intelligencer 2(2) (1980), p. 66. - _Reinhard Zumkeller_, May 24 2009
• Index entries for linear recurrences with constant coefficients, signature (9,-30,50,-45,21,-4).

Crossrefs

Cf. A001580, A001585.

Programs

Magma

[4^n+n^4: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

Mathematica

Table[4^n + n^4, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{9,-30,50,-45,21,-4},{1,5,32,145,512,1649},30] (* Harvey P. Dale, Mar 06 2023 *)

PARI

a(n)=1<<(n+n)+n^4

Formula

G.f.: -(5*x^5 + 38*x^4 + 43*x^3 - 17*x^2 + 4*x - 1) / ((x - 1)^5*(4*x - 1)). - Colin Barker, Jan 01 2013