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A090989 Number of meaningful differential operations of the n-th order on the space R^4.

%I A090989 #34 Jan 15 2025 16:45:00
%S A090989 4,6,8,12,16,24,32,48,64,96,128,192,256,384,512,768,1024,1536,2048,
%T A090989 3072,4096,6144,8192,12288,16384,24576,32768,49152,65536,98304,131072,
%U A090989 196608,262144,393216,524288,786432,1048576,1572864,2097152,3145728,4194304,6291456,8388608
%N A090989 Number of meaningful differential operations of the n-th order on the space R^4.
%H A090989 G. C. Greubel, <a href="/A090989/b090989.txt">Table of n, a(n) for n = 1..1000</a>
%H A090989 Branko 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 A090989 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 A090989 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,2).
%F A090989 a(k+2) = 2*a(k).
%F A090989 a(n) = b(n+3) where b(n) = gcdConv(c(n)) = Sum_{k=0..n} gcd(c(k),c(n-k)) and c(k)=A000079(k) for k>0 and c(0)=1. - _Tilman Neumann_, Jan 11 2009 [Updated by _Sean A. Irvine_, Jan 15 2025]
%F A090989 G.f.: 2*x*(2+3*x)/(1-2*x^2). - _Colin Barker_, May 03 2012
%F A090989 a(n) = 2*A164090(n). - _R. J. Mathar_, Jan 25 2023
%F A090989 a(n) = (sqrt(2))^n*(3/2 + sqrt(2) + (-1)^n*(3/2 - sqrt(2))). - _Taras Goy_, Jan 04 2025
%p A090989 NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 4; # <- 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 A090989 LinearRecurrence[{0,2}, {4,6}, 40] (* _G. C. Greubel_, Feb 02 2019 *)
%o A090989 (PARI) my(x='x+O('x^40)); Vec(2*x*(2+3*x)/(1-2*x^2)) \\ _G. C. Greubel_, Feb 02 2019
%o A090989 (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(2+3*x)/(1-2*x^2) )); // _G. C. Greubel_, Feb 02 2019
%o A090989 (Sage) (2*(2+3*x)/(1-2*x^2)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 02 2019
%o A090989 (GAP) a:=[4,6];; for n in [3..40] do a[n]:=2*a[n-2]; od; a; # _G. C. Greubel_, Feb 02 2019
%Y A090989 Cf. A063759, A029744.
%Y A090989 Cf. A090990, A090991, A090992, A090993, A090994, A090995.
%K A090989 nonn,easy
%O A090989 1,1
%A A090989 _Branko Malesevic_, Feb 29 2004
%E A090989 More terms from _Tilman Neumann_, Feb 06 2009