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 A090993 #32 Sep 08 2022 08:45:12
%S A090993 8,14,24,42,72,126,216,378,648,1134,1944,3402,5832,10206,17496,30618,
%T A090993 52488,91854,157464,275562,472392,826686,1417176,2480058,4251528,
%U A090993 7440174,12754584,22320522,38263752,66961566,114791256,200884698
%N A090993 Number of meaningful differential operations of the n-th order on the space R^8.
%C A090993 Also (starting 5,8,...) the number of zig-zag paths from top to bottom of a rectangle of width 5. - _Joseph Myers_, Dec 23 2008
%C A090993 Number of walks of length n on the path graph P_5. - _Andrew Howroyd_, Apr 17 2017
%H A090993 G. C. Greubel, <a href="/A090993/b090993.txt">Table of n, a(n) for n = 1..1000</a>
%H A090993 B. Malesevic, <a href="https://www.jstor.org/stable/43666958">Some combinatorial aspects of differential operation composition on the space R^n</a>, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
%H A090993 Branko Malesevic, <a href="http://arxiv.org/abs/0704.0750">Some combinatorial aspects of differential operation compositions on space R^n</a>, arXiv:0704.0750 [math.DG], 2007.
%H A090993 Joseph Myers, <a href="http://www.polyomino.org.uk/publications/2008/bmo1-2009-q1.pdf">BMO 2008--2009 Round 1 Problem 1---Generalisation</a>
%H A090993 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,3).
%F A090993 a(n+4) = 4*a(n+2) - 3*a(n).
%F A090993 From _Colin Barker_, May 03 2012: (Start)
%F A090993 a(n) = 3*a(n-2).
%F A090993 G.f.: 2*x*(4+7*x)/(1-3*x^2). (End)
%F A090993 a(n) = (11+3*(-1)^n) * 3^floor((n-1)/2). - _Ralf Stephan_, Jul 19 2013
%p A090993 NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 8; # <- DIMENSION Fun := (i,j)->piecewise(((j=i+1) or (i+j=n+1)),1,0); Identity := (i,j)->piecewise(i=j,1,0); v := matrix(1,n,1); A := piecewise(k>1,(matrix(n,n,Fun))^(k-1),k=1,matrix(n,n,Identity)); return(evalm(v&*A&*transpose(v))[1,1]); end:
%t A090993 LinearRecurrence[{0, 3}, {8, 14}, 32] (* _Jean-François Alcover_, Jul 01 2018 *)
%o A090993 (PARI) my(x='x+O('x^40)); Vec(2*x*(4+7*x)/(1-3*x^2)) \\ _G. C. Greubel_, Feb 02 2019
%o A090993 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 2*x*(4+7*x)/(1-3*x^2) )); // _G. C. Greubel_, Feb 02 2019
%o A090993 (Sage) a=(2*x*(4+7*x)/(1-3*x^2)).series(x, 40).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Feb 02 2019
%o A090993 (GAP) a:=[8,14];; for n in [3..40] do a[n]:=3*a[n-2]; od; a; # _G. C. Greubel_, Feb 02 2019
%Y A090993 Cf. A090989-A090995.
%Y A090993 Column 5 of A220062.
%K A090993 nonn,easy
%O A090993 1,1
%A A090993 _Branko Malesevic_, Feb 29 2004
%E A090993 More terms from _Joseph Myers_, Dec 23 2008