A109808 a(n) = 2*7^(n-1).

2, 14, 98, 686, 4802, 33614, 235298, 1647086, 11529602, 80707214, 564950498, 3954653486, 27682574402, 193778020814, 1356446145698, 9495123019886, 66465861139202, 465261027974414, 3256827195820898, 22797790370746286, 159584532595224002, 1117091728166568014

Offset: 1

Author

Woong Kook (andrewk(AT)math.uri.edu), Aug 16 2005

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000420 (powers of 7), A005277 (nontotients), A132023.

Programs

Magma

[2*7^(n-1):n in [1..25]]; // Vincenzo Librandi, Sep 15 2011

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 *)

PARI

a(n)=7^n*2/7 \\ Charles R Greathouse IV, Jun 10 2011

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)

Extensions

Name changed by Arkadiusz Wesolowski, Oct 20 2013