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, 70368744457505, 281474977042432, 1125899907233249

Offset

0,2

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 + 4*y^4 = (x^2 + 2*x*y + 2*y^2)*(x^2 - 2*x*y + 2*y^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, Vol. 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).

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

From Elmo R. Oliveira, Dec 25 2025: (Start)

E.g.f.: exp(x)*(x*(1 + 7*x + 6*x^2 + x^3) + exp(3*x)).

a(n) = 9*a(n-1) - 30*a(n-2) + 50*a(n-3) - 45*a(n-4) + 21*a(n-5) - 4*a(n-6). (End)

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 *)

Prog

(PARI) a(n)=1<<(n+n)+n^4
(Magma) [4^n+n^4: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

Crossrefs

Cf. A001580, A001585.

Sequence in context: A319790 A273623 A359522 * A271903 A392084 A177467

Adjacent sequences: A001586 A001587 A001588 * A001590 A001591 A001592

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved