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 A123109 #42 Dec 05 2024 13:00:55
%S A123109 1,3,12,39,120,363,1092,3279,9840,29523,88572,265719,797160,2391483,
%T A123109 7174452,21523359,64570080,193710243,581130732,1743392199,5230176600,
%U A123109 15690529803,47071589412,141214768239,423644304720,1270932914163
%N A123109 a(0) = 1, a(1) = 3, a(n) = 3*a(n-1) + 3 for n > 1.
%C A123109 From _R. J. Mathar_, Oct 12 2010: (Start)
%C A123109 The top row, n=2, of an array that counts chess king walks with k >= 0 steps on an n X n board, starting at one of the four corners:
%C A123109 1,3,12, 39,120, 363, 1092, 3279, 9840, 29523, 88572, 265719, 797160,
%C A123109 1,3,21,101,501,2405,11653, 56197, 271493, 1310597, 6328709, 30556549,
%C A123109 1,3,21,126,741,4341,25416,148791, 871041, 5099166,29851041,174751041,
%C A123109 1,3,21,126,810,5169,33447,215796,1395588, 9018255,58302057,376845978,
%C A123109 1,3,21,126,810,5360,36167,246034,1680313,11495503,78705226,539048956,
%C A123109 1,3,21,126,810,5360,36700,254756,1788468,12617828,89338116,633604564,
%C A123109 1,3,21,126,810,5360,36700,256255,1816090,12993280,93566653,676648735,
%C A123109 1,3,21,126,810,5360,36700,256255,1820335,13080120,94845670,692120270,
%C A123109 1,3,21,126,810,5360,36700,256255,1820335,13092211,95117374,696421066,
%C A123109 1,3,21,126,810,5360,36700,256255,1820335,13092211,95151979,697268152,
%C A123109 1,3,21,126,810,5360,36700,256255,1820335,13092211,95151979,697367593,
%C A123109 These are partial sums along rows of the array described in A086346. (End)
%H A123109 Vincenzo Librandi, <a href="/A123109/b123109.txt">Table of n, a(n) for n = 0..1000</a>
%H A123109 Madeleine Goertz and Aaron Williams, <a href="https://arxiv.org/abs/2411.19291">The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles</a>, arXiv:2411.19291 [math.CO], 2024. See p. 17.
%H A123109 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A123109 a(0) = 1 and a(n) = 3*A003462(n) for n > 0.
%F A123109 G.f.: (1-x+3*x^2)/(1-4*x+3*x^2). [Corrected by _Georg Fischer_, May 24 2019]
%F A123109 a(n) = Sum_{k=0..n} 3^k*A123110(n,k). - _Philippe Deléham_, Feb 09 2007
%F A123109 a(n) = A029858(n+1), n > 0. - _R. J. Mathar_, Jun 18 2008
%F A123109 a(n+1) - a(n) = 3^n, n >= 2. - _R. J. Mathar_, Aug 18 2011
%F A123109 E.g.f.: 1 + 3*(exp(3*x) - exp(x))/2. - _G. C. Greubel_, May 24 2019
%t A123109 LinearRecurrence[{4,-3}, {1,3,12}, 30] (* _Georg Fischer_, May 24 2019 *)
%t A123109 Join[{1},NestList[3#+3&,3,30]] (* _Harvey P. Dale_, Aug 16 2020 *)
%o A123109 (Magma) I:=[1, 3, 12]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 27 2012
%o A123109 (PARI) my(x='x+O('x^30)); Vec((1-x+3*x^2)/(1-4*x+3*x^2)) \\ _G. C. Greubel_, May 24 2019
%o A123109 (Sage) ((1-x+3*x^2)/(1-4*x+3*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019
%o A123109 (GAP) a:=[1,3,12];; for n in [4..30] do a[n]:=4*a[n-1]-3*a[n-2]; od; a; # _G. C. Greubel_, May 24 2019
%K A123109 nonn,easy
%O A123109 0,2
%A A123109 _Philippe Deléham_, Sep 28 2006