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, 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);
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
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved