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

7, 13, 24, 45, 84, 158, 296, 557, 1045, 1966, 3691, 6942, 13038, 24516, 46055, 86585, 162680, 305809, 574624, 1080106, 2029680, 3814941, 7169145, 13474502, 25322375, 47592650, 89441626, 168100324, 315917527, 593742597, 1115852904, 2097145317

Offset: 1

Author

Branko Malesevic, Feb 29 2004

Keywords

• nonn
• easy

Comments

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

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

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.
• B. Malesevic and I. Jovovic, The Compositions of the Differential Operations and Gateaux DirectionalDerivative, arXiv:0706.0249 [math.CO], 2007.
• Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
• 2008/9 British Mathematical Olympiad Round 1: Thursday, 4 December 2008, Problem 1 [From _Joseph Myers_, Dec 23 2008]
• Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-1).

Crossrefs

Cf. A090989-A090995.

Partial sums of pairwise sums of A065455.

Cf. A000079, A007283, A020701, A020714, A129638.

Programs

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

Magma

m:=40; R:=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

Maple

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:

Mathematica

LinearRecurrence[{1, 3, -2, -1}, {7, 13, 24, 45}, 32] (* Jean-François Alcover, Nov 25 2017 *)

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

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

Formula

a(n+4) = a(n+3) + 3*a(n+2) - 2*a(n+1) - a(n).

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

Extensions

More terms from Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007

More terms from Joseph Myers, Dec 23 2008