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%I #78 Sep 08 2022 08:44:40
%S 0,1,5,31,185,1111,6665,39991,239945,1439671,8638025,51828151,
%T 310968905,1865813431,11194880585,67169283511,403015701065,
%U 2418094206391,14508565238345,87051391430071,522308348580425
%N a(n) = 5*a(n-1) + 6*a(n-2), a(0) = 0, a(1) = 1.
%C Number of walks of length n between any two distinct vertices of the complete graph K_7. Example: a(2)=5 because the walks of length 2 between the vertices A and B of the complete graph ABCDEFG are ACB, ADB, AEB, AFB and AGB. - _Emeric Deutsch_, Apr 01 2004
%C General form: k=6^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500. -_Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%C Pisano period lengths: 1, 1, 2, 2, 2, 2, 14, 2, 2, 2, 10, 2, 12, 14, 2, 2, 16, 2, 18, 2, ... - _R. J. Mathar_, Aug 10 2012
%C Sum_{i=0..m} (-1)^(m+i)*6^i, for m >= 0, gives all terms after 0. - _Bruno Berselli_, Aug 28 2013
%C The ratio a(n+1)/a(n) converges to 6 as n approaches infinity. Also A053524, A080424, A051958. - _Felix P. Muga II_, Mar 09 2014
%H Vincenzo Librandi, <a href="/A015540/b015540.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 Ji Young Choi, <a href="https://www.emis.de/journals/JIS/VOL21/Choi/choi10.html">A Generalization of Collatz Functions and Jacobsthal Numbers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
%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; Preprint on ResearchGate.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,6).
%F a(n) = 5*a(n-1) + 6*a(n-2).
%F From _Paul Barry_, Apr 20 2003: (Start)
%F a(n) = (6^n - (-1)^n)/7.
%F G.f.: x/((1-6*x)*(1+x)).
%F E.g.f.: (exp(6*x) - exp(-x))/7. (End)
%F a(n) = 6^(n-1) - a(n-1). - _Emeric Deutsch_, Apr 01 2004
%F a(n+1) = Sum_{k=0..n} binomial(n-k, k)*5^(n-2*k)*6^k. - _Paul Barry_, Jul 29 2004
%F a(n) = round(6^n/7). - _Mircea Merca_, Dec 28 2010
%F a(n) = (-1)^(n-1)*Sum_{k=0..n-1} A135278(n-1,k)*(-7)^k) = (6^n - (-1)^n)/7 = (-1)^(n-1)*Sum_{k=0..n-1} (-6)^k. Equals (-1)^(n-1)*Phi(n,-6), where Phi is the cyclotomic polynomial when n is an odd prime. (For n > 0.) - _Tom Copeland_, Apr 14 2014
%e G.f. = x + 5*x^2 + 31*x^3 + 185*x^4 + 1111*x^5 + 6665*x^6 + 39991*x^7 + ...
%p seq(round(6^n/7),n=0..25); # _Mircea Merca_, Dec 28 2010
%t k=0; lst={k}; Do[k = 6^n-k; AppendTo[lst, k], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008 *)
%t CoefficientList[Series[x / ((1 - 6 x) (1 + x)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 26 2014 *)
%t LinearRecurrence[{5,6},{0,1},30] (* _Harvey P. Dale_, May 12 2015 *)
%o (Sage) [lucas_number1(n,5,-6) for n in range(21)] # _Zerinvary Lajos_, Apr 24 2009
%o (Magma) [Floor(6^n/7-(-1)^n/7): n in [0..30]]; // _Vincenzo Librandi_, Jun 24 2011
%o (PARI) x='x+O('x^30); concat([0], Vec(x/((1-6*x)*(1+x)))) \\ _G. C. Greubel_, Jan 24 2018
%o (PARI) a(n) = round(6^n/7); \\ _Altug Alkan_, Sep 05 2018
%Y Partial sums are in A033116. Cf. A014987.
%K nonn,easy
%O 0,3
%A _Olivier Gérard_
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