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

0,1

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).

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)

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 *)

Prog

(Magma) [4^n+1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011
(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
(GAP) List([0..30], n-> 4^n+1) # G. C. Greubel, May 09 2019

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.

Sequence in context: A090902 A150012 A150013 * A123166 A008932 A167809

Adjacent sequences: A052536 A052537 A052538 * A052540 A052541 A052542

Keyword

easy,nonn

Author

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

Status

approved