%I #83 Sep 08 2022 08:44:59

%S 2,5,17,65,257,1025,4097,16385,65537,262145,1048577,4194305,16777217,

%T 67108865,268435457,1073741825,4294967297,17179869185,68719476737,

%U 274877906945,1099511627777,4398046511105,17592186044417

%N a(n) = 4^n + 1.

%C 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.

%C Also the edge cover number of the (n+1)-Sierpinski tetrahedron graph. - _Eric W. Weisstein_, Sep 20 2017

%C First bisection of A000051, A049332, A052531 and A014551. - _Klaus Purath_, Sep 23 2020

%H Vincenzo Librandi, <a href="/A052539/b052539.txt">Table of n, a(n) for n = 0..175</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=470">Encyclopedia of Combinatorial Structures 470</a>.

%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].

%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.18483/ijSci.2188">Some Groupoids and their Representations by Means of Integer Sequences</a>, International Journal of Sciences 8(10) (2019).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCoverNumber.html">Edge Cover Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiTetrahedronGraph.html">Sierpinski Tetrahedron Graph</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).

%F a(n) = 4^n + 1.

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

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

%F E.g.f.: exp(x) + exp(4*x). - _Mohammad K. Azarian_, Jan 02 2009

%F From _Klaus Purath_, Sep 23 2020: (Start)

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

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

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

%p spec := [S,{S=Union(Sequence(Union(Z,Z,Z,Z)),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);

%p A052539:=n->4^n + 1; seq(A052539(n), n=0..30); # _Wesley Ivan Hurt_, Jun 12 2014

%t Table[4^n + 1, {n, 0, 30}]

%t (* From _Eric W. Weisstein_, Sep 20 2017 *)

%t 4^Range[0, 30] + 1

%t LinearRecurrence[{5, -4}, {2, 5}, 30]

%t CoefficientList[Series[(2-5x)/(1-5x+4x^2), {x, 0, 30}], x] (* End *)

%o (Magma) [4^n+1: n in [0..30] ]; // _Vincenzo Librandi_, Apr 30 2011

%o (PARI) a(n)=4^n+1 \\ _Charles R Greathouse IV_, Nov 20 2011

%o (Sage) [4^n+1 for n in (0..30)] # _G. C. Greubel_, May 09 2019

%o (GAP) List([0..30], n-> 4^n+1) # _G. C. Greubel_, May 09 2019

%Y Cf. A164346 (first differences).

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

%Y Cf. A019434, A274903.

%K easy,nonn

%O 0,1

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