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a(n) = 7*a(n-1) + 8*a(n-2), a(0) = 0, a(1) = 1.
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%I #61 Aug 17 2024 22:31:15

%S 0,1,7,57,455,3641,29127,233017,1864135,14913081,119304647,954437177,

%T 7635497415,61083979321,488671834567,3909374676537,31274997412295,

%U 250199979298361,2001599834386887,16012798675095097,128102389400760775,1024819115206086201,8198552921648689607

%N a(n) = 7*a(n-1) + 8*a(n-2), a(0) = 0, a(1) = 1.

%C A linear 2nd order recurrence. A Jacobsthal number sequence.

%C Binomial transform of A053573 (preceded by zero). - _Paul Barry_, Apr 09 2003

%C Second binomial transform of A080424. Binomial transform of A053573, with leading zero. Binomial transform is 0,1,9,81,729,....(9^n - 0^n)/9. Second binomial transform is 0,1,11,111,1111,... (A002275: repunits). - _Paul Barry_, Mar 14 2004

%C Number of walks of length n between any two distinct nodes of the complete graph K_9. Example: a(2)=7 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHI are: ACB, ADB, AEB, AFB, AGB, AHB and AIB. - _Emeric Deutsch_, Apr 01 2004

%C Unsigned version of A014990. - _Philippe Deléham_, Feb 13 2007

%C General form: k=8^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008

%C The ratio a(n+1)/a(n) converges to 8 as n approaches infinity. - _Felix P. Muga II_, Mar 09 2014

%H Vincenzo Librandi, <a href="/A015565/b015565.txt">Table of n, a(n) for n = 0..1000</a>

%H Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, and 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 M. Dukes and C. D. White, <a href="http://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016.

%H Dale Gerdemann, <a href="https://www.youtube.com/watch?v=WJBOPr9l9iY">Fractal generated from (7,8) recursion</a>, YouTube Video, Dec 5, 2014.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,8).

%F From _Paul Barry_, Apr 09 2003: (Start)

%F a(n) = (8^n - (-1)^n)/9.

%F a(n) = J(3*n)/3 = A001045(3*n)/3. (End)

%F From _Emeric Deutsch_, Apr 01 2004: (Start)

%F a(n) = 8^(n-1) - a(n-1).

%F G.f.: x/(1-7*x-8*x^2). (End)

%F a(n) = Sum_{k = 0..n} A106566(n,k)*A099322(k). - _Philippe Deléham_, Oct 30 2008

%F a(n) = round(8^n/9). - _Mircea Merca_, Dec 28 2010

%F From _Peter Bala_, May 31 2024: (Start)

%F G.f: A(x) = x/(1 - x^2) o x/(1 - x^2), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A054878.

%F The black diamond product A(x) o A(x) is the g.f. for the number of walks of length n between any two distinct nodes of the complete graph K_81.

%F Row 8 of A062160. (End)

%F E.g.f.: exp(-x)*(exp(9*x) - 1)/9. - _Elmo R. Oliveira_, Aug 17 2024

%e G.f. = x + 7*x^2 + 57*x^3 + 455*x^4 + 3641*x^5 + 29127*x^6 + 233017*x^7 + ...

%p seq(round(8^n/9),n=0..25); # _Mircea Merca_, Dec 28 2010

%t k=0;lst={k};Do[k=8^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008 *)

%t LinearRecurrence[{7,8},{0,1},30] (* _Harvey P. Dale_, Mar 04 2016 *)

%o (Sage) [lucas_number1(n,7,-8) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 24 2009

%o (Magma) [Round(8^n/9): n in [0..30]]; // _Vincenzo Librandi_, Jun 24 2011

%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1-7*x-8*x^2))) \\ _G. C. Greubel_, Dec 30 2017

%Y Cf. A002275, A014990, A053573, A080424, A082311, A082365.

%Y Cf. A062160, A099322, A106566.

%Y Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008

%K nonn,easy

%O 0,3

%A _Olivier Gérard_