cp's OEIS Frontend

A090991 Number of meaningful differential operations of the n-th order on the space R^6.

%I A090991 #45 Aug 27 2025 12:03:35
%S A090991 6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362,13530,
%T A090991 21892,35422,57314,92736,150050,242786,392836,635622,1028458,1664080,
%U A090991 2692538,4356618,7049156,11405774,18454930,29860704,48315634,78176338,126491972
%N A090991 Number of meaningful differential operations of the n-th order on the space R^6.
%C A090991 Apparently a(n) = A054886(n+2) for n=1..1000. - _Georg Fischer_, Oct 06 2018
%H A090991 Indranil Ghosh, <a href="/A090991/b090991.txt">Table of n, a(n) for n = 1..4772</a>
%H A090991 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A090991 Branko J. 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 A090991 Branko J. 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 A090991 Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, <a href="https://ceur-ws.org/Vol-3792/paper19.pdf">Integer sequences from k-iterated line digraphs</a>, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
%H A090991 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A090991 a(k+4) = 3*a(k+2) - a(k).
%F A090991 a(k) = 2*Fibonacci(k+3).
%F A090991 From _Philippe Deléham_, Nov 19 2008: (Start)
%F A090991 a(n) = a(n-1) + a(n-2), n>2, where a(1)=6, a(2)=10.
%F A090991 G.f.: 2*x*(3+2*x)/(1-x-x^2). (End)
%F A090991 E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 4. - _Stefano Spezia_, Apr 18 2022
%p A090991 NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 6; # <- 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 A090991 CoefficientList[Series[2*(3+2z)/(1-z-z^2), {z, 0, 40}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 11 2011 *)
%o A090991 (GAP) a:=[6,10];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # _Muniru A Asiru_, Oct 06 2018
%o A090991 (PARI) my(x='x+O('x^40)); Vec(2*x*(3+2*x)/(1-x-x^2)) \\ _G. C. Greubel_, Feb 02 2019
%o A090991 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(3+2*x)/(1-x-x^2) )); // _G. C. Greubel_, Feb 02 2019
%o A090991 (Sage) (2*(3+2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 02 2019
%Y A090991 Cf. A054886, A055389, A068922, A090989-A090995.
%Y A090991 Essentially the same as A006355, A047992 and A078642.
%K A090991 nonn,easy,changed
%O A090991 1,1
%A A090991 _Branko Malesevic_, Feb 29 2004