%I #46 Aug 17 2024 20:40:14
%S 0,1,6,43,300,2101,14706,102943,720600,5044201,35309406,247165843,
%T 1730160900,12111126301,84777884106,593445188743,4154116321200,
%U 29078814248401,203551699738806,1424861898171643,9974033287201500,69818233010410501,488727631072873506
%N a(n) = 6*a(n-1) + 7*a(n-2), a(0) = 0, a(1) = 1.
%C Number of walks of length n between any two distinct nodes of the complete graph K_8. Example: a(2)=6 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGH are: ACB, ADB, AEB, AFB, AGB and AHB. - _Emeric Deutsch_, Apr 01 2004
%C General form: k=7^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%C The ratio a(n+1)/a(n) converges to 7 as n approaches infinity. - _Felix P. Muga II_, Mar 09 2014
%H Vincenzo Librandi, <a href="/A015552/b015552.txt">Table of n, a(n) for n = 0..1000</a>
%H Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, Jiemeng Zhang, <a href="https://arxiv.org/abs/1911.01687">Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences</a>, arXiv:1911.01687 [math.CO], 2019.
%H F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,7).
%F a(n) = 6*a(n-1) + 7*a(n-2).
%F From _Emeric Deutsch_, Apr 01 2004: (Start)
%F G.f.: x/(1-6*x-7*x^2).
%F a(n) = 7^(n-1) - a(n-1). (End)
%F a(n) = (7^n - (-1)^n)/8. - _Rolf Pleisch_, Jul 06 2009
%F a(n) = round(7^n/8). - _Mircea Merca_, Dec 28 2010
%F E.g.f. exp(3*x)*sinh(4*x)/4. - _Elmo R. Oliveira_, Aug 17 2024
%e G.f. = x + 6*x^2 + 43*x^3 + 300*x^4 + 2101*x^5 + 14706*x^6 + 102943*x^7 + ...
%p seq(round(7^n/8),n=0..25); # _Mircea Merca_, Dec 28 2010
%t k=0;lst={k};Do[k=7^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008 *)
%t Table[(7^n - (-1)^n)/8, {n,0,30}] (* _G. C. Greubel_, Dec 2017 *)
%o (PARI) {a(n) = if ( n<0, 0, (7^n - (-1)^n) / 8)};
%o (Sage) [lucas_number1(n,6,-7) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 24 2009
%o (Magma) [Round(7^n/8): n in [0..30]]; // _Vincenzo Librandi_, Jun 24 2011
%Y Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%K nonn,easy
%O 0,3
%A _Olivier Gérard_