A090993 Number of meaningful differential operations of the n-th order on the space R^8.

8, 14, 24, 42, 72, 126, 216, 378, 648, 1134, 1944, 3402, 5832, 10206, 17496, 30618, 52488, 91854, 157464, 275562, 472392, 826686, 1417176, 2480058, 4251528, 7440174, 12754584, 22320522, 38263752, 66961566, 114791256, 200884698

Offset: 1

Author

Branko Malesevic, Feb 29 2004

Keywords

• nonn
• easy

Comments

Also (starting 5,8,...) the number of zig-zag paths from top to bottom of a rectangle of width 5. - Joseph Myers, Dec 23 2008

Number of walks of length n on the path graph P_5. - Andrew Howroyd, Apr 17 2017

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• B. 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.
• Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
• Index entries for linear recurrences with constant coefficients, signature (0,3).

Crossrefs

Cf. A090989-A090995.

Column 5 of A220062.

Programs

GAP

a:=[8,14];; for n in [3..40] do a[n]:=3*a[n-2]; od; a; # G. C. Greubel, Feb 02 2019

Magma

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  2*x*(4+7*x)/(1-3*x^2) )); // G. C. Greubel, Feb 02 2019

Maple

NUM := proc(k :: integer) local i,j,n,Fun,Identity,v,A; n := 8; # <- 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:

Mathematica

LinearRecurrence[{0, 3}, {8, 14}, 32] (* Jean-François Alcover, Jul 01 2018 *)

PARI

my(x='x+O('x^40)); Vec(2*x*(4+7*x)/(1-3*x^2)) \\ G. C. Greubel, Feb 02 2019

Sage

a=(2*x*(4+7*x)/(1-3*x^2)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019

Formula

a(n+4) = 4*a(n+2) - 3*a(n).

From Colin Barker, May 03 2012: (Start)

a(n) = 3*a(n-2).

G.f.: 2*x*(4+7*x)/(1-3*x^2). (End)

a(n) = (11+3*(-1)^n) * 3^floor((n-1)/2). - Ralf Stephan, Jul 19 2013

Extensions

More terms from Joseph Myers, Dec 23 2008