A052539 a(n) = 4^n + 1.

2, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577, 4194305, 16777217, 67108865, 268435457, 1073741825, 4294967297, 17179869185, 68719476737, 274877906945, 1099511627777, 4398046511105, 17592186044417

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

The sequence is a Lucas sequence V(P,Q) with P = 5 and Q = 4, so if n is a prime number, then V_n(5,4) - 5 is divisible by n. The smallest pseudoprime q which divides V_q(5,4) - 5 is 15.

Also the edge cover number of the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 20 2017

First bisection of A000051, A049332, A052531 and A014551. - Klaus Purath, Sep 23 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..175
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 470.
• Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
• Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences 8(10) (2019).
• Eric Weisstein's World of Mathematics, Edge Cover Number.
• Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
• Wikipedia, Lucas sequence: Specific names.
• Index entries for linear recurrences with constant coefficients, signature (5,-4).

Crossrefs

Cf. A164346 (first differences).

Other powers: A000051, A034472, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081.

Cf. A019434, A274903.

Programs

GAP

List([0..30], n-> 4^n+1); # G. C. Greubel, May 09 2019

Magma

[4^n+1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011

Maple

spec := [S,{S=Union(Sequence(Union(Z,Z,Z,Z)),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);
A052539:=n->4^n + 1; seq(A052539(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014

Mathematica

Table[4^n + 1, {n, 0, 30}]
(* From Eric W. Weisstein, Sep 20 2017 *)
4^Range[0, 30] + 1
LinearRecurrence[{5, -4}, {2, 5}, 30]
CoefficientList[Series[(2-5x)/(1-5x+4x^2), {x, 0, 30}], x] (* End *)

PARI

a(n)=4^n+1 \\ Charles R Greathouse IV, Nov 20 2011

Sage

[4^n+1 for n in (0..30)] # G. C. Greubel, May 09 2019

Formula

a(n) = 4^n + 1.

a(n) = 4*a(n-1) - 3 = 5*a(n-1) - 4*a(n-2).

G.f.: (2 - 5*x)/((1 - 4*x)*(1 - x)).

E.g.f.: exp(x) + exp(4*x). - Mohammad K. Azarian, Jan 02 2009

From Klaus Purath, Sep 23 2020: (Start)

a(n) = 3*4^(n-1) + a(n-1).

a(n) = (a(n-1)^2 + 9*4^(n-2))/a(n-2).

a(n) = A178675(n) - 3. (End)