|
%I #56 May 08 2023 02:26:27
%S 2,14,98,686,4802,33614,235298,1647086,11529602,80707214,564950498,
%T 3954653486,27682574402,193778020814,1356446145698,9495123019886,
%U 66465861139202,465261027974414,3256827195820898,22797790370746286,159584532595224002,1117091728166568014
%N a(n) = 2*7^(n-1).
%C 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).
%C 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.
%C 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
%C Apart from first term 2, these are the numbers that satisfy phi(n) = 3*n/7. - _Michel Marcus_, Jul 14 2015
%H Vincenzo Librandi, <a href="/A109808/b109808.txt">Table of n, a(n) for n = 1..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H W. Kook, <a href="http://dx.doi.org/10.1016/j.dam.2006.11.002">Edge-rooted forests and alpha-invariant of cone graphs</a>, Discrete Applied Mathematics, Volume 155, Issue 8, 15 April 2007, Pages 1071-1075.
%H Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, <a href="http://arxiv.org/abs/1511.00080">Pattern Avoidance in Task-Precedence Posets</a>, arXiv:1511.00080 [math.CO], 2015-2016.
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (7).
%F a(n) = 2*7^(n-1); a(n) = 7*a(n-1) where a(1) = 2.
%F G.f.: 2*x/(1 - 7*x). - _Philippe Deléham_, Nov 23 2008
%F E.g.f.: 2*(exp(7*x) - 1)/7. - _Stefano Spezia_, May 29 2021
%F From _Amiram Eldar_, May 08 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 7/12.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 7/16.
%F Product_{n>=1} (1 - 1/a(n)) = A132023. (End)
%p a:= n-> 2*7^(n-1): seq(a(n), n=1..30);
%t 2*7^Range[0, 40] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)
%o (PARI) a(n)=7^n*2/7 \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Magma) [2*7^(n-1):n in [1..25]]; // _Vincenzo Librandi_, Sep 15 2011
%Y Cf. A000420 (powers of 7), A005277 (nontotients), A132023.
%K nonn,easy
%O 1,1
%A Woong Kook (andrewk(AT)math.uri.edu), Aug 16 2005
%E Name changed by _Arkadiusz Wesolowski_, Oct 20 2013
| 