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%I #31 Oct 21 2022 21:59:43
%S 10,18,32,58,104,188,338,610,1098,1980,3566,6428,11580,20870,37602,
%T 67762,122096,220018,396448,714388,1287266,2319594,4179738,7531660,
%U 13571542,24455124,44066548,79405254,143083226,257827186,464588384
%N Number of meaningful differential operations of the n-th order on the space R^10.
%C Also (starting 6,10,...) the number of zig-zag paths from top to bottom of a rectangle of width 6. - _Joseph Myers_, Dec 23 2008
%C Number of walks of length n on the path graph P_6. - _Andrew Howroyd_, Apr 17 2017
%H G. C. Greubel, <a href="/A090995/b090995.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 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 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1).
%F Equals 2 * A090990.
%F a(k+6) = 5*a(k+4) - 6*a(k+2) + a(k).
%F From _Colin Barker_, May 03 2012: (Start)
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3).
%F G.f.: 2*x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3). (End)
%p NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 10; # <- 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 a[n_ /; n <= 6] := {10, 18, 32, 58, 104, 188}[[n]]; a[n_] := a[n] = 5*a[n-2] - 6*a[n-4] + a[n-6]; Array[a, 31] (* _Jean-François Alcover_, Oct 07 2017 *)
%t 2*LinearRecurrence[{1,2,-1}, {5,9,16}, 40] (* _G. C. Greubel_, Feb 02 2019 *)
%o (PARI) my(x='x+O('x^40)); Vec(2*x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3)) \\ _G. C. Greubel_, Feb 02 2019
%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 2*x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3) )); // _G. C. Greubel_, Feb 02 2019
%o (Sage) a=(2*x*(5+4*x-3*x^2)/(1-x-2*x^2+x^3)).series(x, 40).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Feb 02 2019
%o (GAP) a:=[10,18,32];; for n in [4..30] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, Feb 02 2019
%Y Cf. A090989-A090994.
%Y Column 6 of A220062.
%K nonn,easy
%O 1,1
%A _Branko Malesevic_, Feb 29 2004
%E More terms from _Joseph Myers_, Dec 23 2008
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