This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A015565 #61 Aug 17 2024 22:31:15
%S A015565 0,1,7,57,455,3641,29127,233017,1864135,14913081,119304647,954437177,
%T A015565 7635497415,61083979321,488671834567,3909374676537,31274997412295,
%U A015565 250199979298361,2001599834386887,16012798675095097,128102389400760775,1024819115206086201,8198552921648689607
%N A015565 a(n) = 7*a(n-1) + 8*a(n-2), a(0) = 0, a(1) = 1.
%C A015565 A linear 2nd order recurrence. A Jacobsthal number sequence.
%C A015565 Binomial transform of A053573 (preceded by zero). - _Paul Barry_, Apr 09 2003
%C A015565 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 A015565 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 A015565 Unsigned version of A014990. - _Philippe Deléham_, Feb 13 2007
%C A015565 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 A015565 The ratio a(n+1)/a(n) converges to 8 as n approaches infinity. - _Felix P. Muga II_, Mar 09 2014
%H A015565 Vincenzo Librandi, <a href="/A015565/b015565.txt">Table of n, a(n) for n = 0..1000</a>
%H A015565 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 A015565 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 A015565 Dale Gerdemann, <a href="https://www.youtube.com/watch?v=WJBOPr9l9iY">Fractal generated from (7,8) recursion</a>, YouTube Video, Dec 5, 2014.
%H A015565 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,8).
%F A015565 From _Paul Barry_, Apr 09 2003: (Start)
%F A015565 a(n) = (8^n - (-1)^n)/9.
%F A015565 a(n) = J(3*n)/3 = A001045(3*n)/3. (End)
%F A015565 From _Emeric Deutsch_, Apr 01 2004: (Start)
%F A015565 a(n) = 8^(n-1) - a(n-1).
%F A015565 G.f.: x/(1-7*x-8*x^2). (End)
%F A015565 a(n) = Sum_{k = 0..n} A106566(n,k)*A099322(k). - _Philippe Deléham_, Oct 30 2008
%F A015565 a(n) = round(8^n/9). - _Mircea Merca_, Dec 28 2010
%F A015565 From _Peter Bala_, May 31 2024: (Start)
%F A015565 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 A015565 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 A015565 Row 8 of A062160. (End)
%F A015565 E.g.f.: exp(-x)*(exp(9*x) - 1)/9. - _Elmo R. Oliveira_, Aug 17 2024
%e A015565 G.f. = x + 7*x^2 + 57*x^3 + 455*x^4 + 3641*x^5 + 29127*x^6 + 233017*x^7 + ...
%p A015565 seq(round(8^n/9),n=0..25); # _Mircea Merca_, Dec 28 2010
%t A015565 k=0;lst={k};Do[k=8^n-k;AppendTo[lst, k], {n, 0, 5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008 *)
%t A015565 LinearRecurrence[{7,8},{0,1},30] (* _Harvey P. Dale_, Mar 04 2016 *)
%o A015565 (Sage) [lucas_number1(n,7,-8) for n in range(0, 21)] # _Zerinvary Lajos_, Apr 24 2009
%o A015565 (Magma) [Round(8^n/9): n in [0..30]]; // _Vincenzo Librandi_, Jun 24 2011
%o A015565 (PARI) x='x+O('x^30); concat([0], Vec(x/(1-7*x-8*x^2))) \\ _G. C. Greubel_, Dec 30 2017
%Y A015565 Cf. A002275, A014990, A053573, A080424, A082311, A082365.
%Y A015565 Cf. A062160, A099322, A106566.
%Y A015565 Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - _Vladimir Joseph Stephan Orlovsky_, Dec 11 2008
%K A015565 nonn,easy
%O A015565 0,3
%A A015565 _Olivier Gérard_