A090989 Number of meaningful differential operations of the n-th order on the space R^4.
4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728, 4194304, 6291456, 8388608
Offset: 1
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Branko Malesevic, Some combinatorial aspects of differential operation composition on the space R^n , Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
- Branko Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
- Index entries for linear recurrences with constant coefficients, signature (0,2).
Programs
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GAP
a:=[4,6];; for n in [3..40] do a[n]:=2*a[n-2]; od; a; # G. C. Greubel, Feb 02 2019
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Magma
m:=40; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( 2*x*(2+3*x)/(1-2*x^2) )); // G. C. Greubel, Feb 02 2019 -
Maple
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:
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Mathematica
LinearRecurrence[{0,2}, {4,6}, 40] (* G. C. Greubel, Feb 02 2019 *) -
PARI
my(x='x+O('x^40)); Vec(2*x*(2+3*x)/(1-2*x^2)) \\ G. C. Greubel, Feb 02 2019 -
Sage
(2*(2+3*x)/(1-2*x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 02 2019
Formula
a(k+2) = 2*a(k).
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]
G.f.: 2*x*(2+3*x)/(1-2*x^2). - Colin Barker, May 03 2012
a(n) = 2*A164090(n). - R. J. Mathar, Jan 25 2023
a(n) = (sqrt(2))^n*(3/2 + sqrt(2) + (-1)^n*(3/2 - sqrt(2))). - Taras Goy, Jan 04 2025
Extensions
More terms from Tilman Neumann, Feb 06 2009