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A052539 a(n) = 4^n + 1. 72
2, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577, 4194305, 16777217, 67108865, 268435457, 1073741825, 4294967297, 17179869185, 68719476737, 274877906945, 1099511627777, 4398046511105, 17592186044417 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..175

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 470

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

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

Sequence in context: A150012 A150013 A123166 * A008932 A167809 A262449

Adjacent sequences:  A052536 A052537 A052538 * A052540 A052541 A052542

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified October 13 18:57 EDT 2019. Contains 327981 sequences. (Running on oeis4.)