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 A077937 #28 Jun 05 2025 20:37:48
%S A077937 1,2,6,14,36,88,220,544,1352,3352,8320,20640,51216,127072,315296,
%T A077937 782304,1941056,4816128,11949760,29649664,73566592,182532992,
%U A077937 452899840,1123732480,2788198656,6918062592,17165057536,42589842944,105673675776,262196922368
%N A077937 Expansion of 1/(1-2*x-2*x^2+2*x^3).
%C A077937 Form the graph with matrix A = [1,1,1,1; 1,0,0,0; 1,0,0,0; 1,0,0,1]. Then the sequence 0, 1, 2, 6, ... counts walks of length n between the degree 5 vertex and the degree 3 vertex. - _Paul Barry_, Oct 02 2004
%C A077937 From _Sean A. Irvine_, Jun 05 2025: (Start)
%C A077937 Also, the number of walks of length n starting at vertex 0 in the graph:
%C A077937 1-2
%C A077937 /| |
%C A077937 0 | |
%C A077937 \| |
%C A077937 4-3. (End)
%H A077937 G. C. Greubel, <a href="/A077937/b077937.txt">Table of n, a(n) for n = 0..1000</a>
%H A077937 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A077937 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2).
%F A077937 a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) with a(0) = 1, a(1) = 2, and a(3) = 8. - _G. C. Greubel_, May 02 2022
%t A077937 LinearRecurrence[{2,2,-2}, {1,2,6}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Jul 03 2011 *)
%t A077937 CoefficientList[Series[1/(1-2*x-2*x^2+2*x^3),{x,0,40}],x] (* _Harvey P. Dale_, Dec 05 2018 *)
%o A077937 (PARI) Vec(1/(1-2*x-2*x^2+2*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A077937 (Magma) [n le 3 select Factorial(n) else 2*(Self(n-1) +Self(n-2) -Self(n-3)): n in [1..51]]; // _G. C. Greubel_, May 02 2022
%o A077937 (SageMath)
%o A077937 def A077937_list(prec):
%o A077937 P.<x> = PowerSeriesRing(ZZ, prec)
%o A077937 return P( 1/(1-2*x-2*x^2+2*x^3) ).list()
%o A077937 A077937_list(50) # _G. C. Greubel_, May 02 2022
%Y A077937 Cf. A052528, A052987, A107300.
%K A077937 nonn,easy
%O A077937 0,2
%A A077937 _N. J. A. Sloane_, Nov 17 2002