cp's OEIS Frontend

A090992 Number of meaningful differential operations of the n-th order on the space R^7.

%I A090992 #31 Sep 08 2022 08:45:12
%S A090992 7,13,24,45,84,158,296,557,1045,1966,3691,6942,13038,24516,46055,
%T A090992 86585,162680,305809,574624,1080106,2029680,3814941,7169145,13474502,
%U A090992 25322375,47592650,89441626,168100324,315917527,593742597,1115852904,2097145317
%N A090992 Number of meaningful differential operations of the n-th order on the space R^7.
%C A090992 Also number of meaningful compositions of the n-th order of the differential operations and Gateaux directional derivative on the space R^6. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007
%C A090992 Also (starting 4,7,...) the number of zig-zag paths from top to bottom of a rectangle of width 8, whose color is that of the top right corner. - _Joseph Myers_, Dec 23 2008
%H A090992 G. C. Greubel, <a href="/A090992/b090992.txt">Table of n, a(n) for n = 1..1000</a>
%H A090992 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 A090992 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 A090992 B. Malesevic and I. Jovovic, <a href="http://arXiv.org/abs/0706.0249">The Compositions of the Differential Operations and Gateaux DirectionalDerivative</a>, arXiv:0706.0249 [math.CO], 2007.
%H A090992 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 A090992 <a href="http://www.bmoc.maths.org/home/bmo1-2009.pdf">2008/9 British Mathematical Olympiad Round 1: Thursday, 4 December 2008</a>, Problem 1 [From _Joseph Myers_, Dec 23 2008]
%H A090992 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-2,-1).
%F A090992 a(n+4) = a(n+3) + 3*a(n+2) - 2*a(n+1) - a(n).
%F A090992 G.f.: x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3)). - _Colin Barker_, Mar 08 2012
%p A090992 NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 7; # <- 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 A090992 LinearRecurrence[{1, 3, -2, -1}, {7, 13, 24, 45}, 32] (* _Jean-François Alcover_, Nov 25 2017 *)
%o A090992 (PARI) my(x='x+O('x^40)); Vec(x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3))) \\ _G. C. Greubel_, Feb 02 2019
%o A090992 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(  x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3)) )); // _G. C. Greubel_, Feb 02 2019
%o A090992 (Sage) a=(x*(7+6*x-10*x^2-4*x^3)/((1-x)*(1-3*x^2-x^3))).series(x, 40).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, Feb 02 2019
%o A090992 (GAP) a:=[7,13,24,45];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2] - 2*a[n-3] - a[n-4]; od; a; # _G. C. Greubel_, Feb 02 2019
%Y A090992 Cf. A090989-A090995.
%Y A090992 Partial sums of pairwise sums of A065455.
%Y A090992 Cf. A000079, A007283, A020701, A020714, A129638.
%K A090992 nonn,easy
%O A090992 1,1
%A A090992 _Branko Malesevic_, Feb 29 2004
%E A090992 More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007
%E A090992 More terms from _Joseph Myers_, Dec 23 2008