OFFSET
1,1
COMMENTS
Value of Tutte dichromatic polynomial T_G(0,1) where G is the Cartesian product of the paths P_2 and P_n (n>1).
The value of Tutte dichromatic polynomial T_G(0,1) where G is the Cartesian product of the paths P_1 and P_n (n>1) is seen to be 2^(n-1), which is also the number of edge-rooted forests in P_n.
In 1956, Andrzej Schinzel showed that for every n >= 2, a(n) is not a value of Euler's function. - Arkadiusz Wesolowski, Oct 20 2013
Apart from first term 2, these are the numbers that satisfy phi(n) = 3*n/7. - Michel Marcus, Jul 14 2015
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences.
W. Kook, Edge-rooted forests and alpha-invariant of cone graphs, Discrete Applied Mathematics, Volume 155, Issue 8, 15 April 2007, Pages 1071-1075.
Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Index entries for linear recurrences with constant coefficients, signature (7).
FORMULA
a(n) = 2*7^(n-1); a(n) = 7*a(n-1) where a(1) = 2.
G.f.: 2*x/(1 - 7*x). - Philippe Deléham, Nov 23 2008
E.g.f.: 2*(exp(7*x) - 1)/7. - Stefano Spezia, May 29 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=1} 1/a(n) = 7/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7/16.
Product_{n>=1} (1 - 1/a(n)) = A132023. (End)
MAPLE
a:= n-> 2*7^(n-1): seq(a(n), n=1..30);
MATHEMATICA
2*7^Range[0, 40] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
PROG
(PARI) a(n)=7^n*2/7 \\ Charles R Greathouse IV, Jun 10 2011
(Magma) [2*7^(n-1):n in [1..25]]; // Vincenzo Librandi, Sep 15 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Woong Kook (andrewk(AT)math.uri.edu), Aug 16 2005
EXTENSIONS
Name changed by Arkadiusz Wesolowski, Oct 20 2013
STATUS
approved