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 A215404 #57 Sep 08 2022 08:46:03
%S A215404 0,0,1,4,13,39,113,322,910,2561,7192,20175,56563,158535,444276,
%T A215404 1244936,3488381,9774440,27387681,76739023,215018609,602469686,
%U A215404 1688083894,4729907909,13252910268,37133833451,104046695091,291532369743,816855560248,2288778436672,6413014696201
%N A215404 a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3), with a(0)=0, a(1)=0 and a(2)=1.
%C A215404 We have a(n)=C(n;-1), A121449(n)=A(n;-1), A085810(n+1)=-B(n+1;-1), where A(n;d), B(n;d), and C(n;d), n in N, d in C, are so-called quasi-Fibonacci numbers defined and discussed in the comments to A121449 and in Witula-Slota-Warzynski's paper. It follows from formulas (3.47-49) in this paper that the value of A(n;1/3), B(n;1/3) and C(n;1/3) could be obtained from special convolution type identities for sequences a(n), A121449, and A085810.
%H A215404 G. C. Greubel, <a href="/A215404/b215404.txt">Table of n, a(n) for n = 0..1000</a>
%H A215404 Ilya Amburg, Krishna Dasaratha, Laure Flapan, Thomas Garrity, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse, and Matthew Stoffregen, <a href="http://arxiv.org/abs/1509.05239">Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences</a>, arXiv:1509.05239 [math.CO], 2015.
%H A215404 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H A215404 Roman Witula, Damian Slota and Adam Warzynski, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Slota/slota57.html">Quasi-Fibonacci Numbers of the Seventh Order</a>, J. Integer Seq., 9 (2006), Article 06.4.3.
%H A215404 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-1).
%F A215404 G.f.: x^2/(1-4*x+3*x^2+x^3).
%F A215404 a(n) = (1/7)*((c(2)-c(4))*(1-c(1))^n + (c(4)-c(1))*(1-c(2))^n + (c(1)-c(2))*(1-c(4))^n), where c(j):=2*cos(2*Pi*j/7) - this formula is the Binet formula for a(n) (see the Binet formula (3.17) for the respective quasi-Fibonacci number C(n;d) for value d=-1 in the Witula-Slota-Warzynski paper).
%t A215404 LinearRecurrence[{4,-3,-1}, {0,0,1}, 50]
%t A215404 CoefficientList[Series[x^2/(1 - 4 x + 3 x^2 + x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 18 2015 *)
%o A215404 (PARI) Vec(x^2/(1-4*x+3*x^2+x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Oct 01 2012
%o A215404 (Magma) I:=[0,0,1]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..35]]; // _Vincenzo Librandi_, Sep 18 2015
%Y A215404 Cf. A077998, A006054, A121449, A085810, A052975, A094789, A005021, A121442.
%K A215404 nonn,easy
%O A215404 0,4
%A A215404 _Roman Witula_, Aug 09 2012