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

1,1

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).

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

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 *)

Prog

(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
(Magma) m:=40; R<x>:=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
(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
(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

Crossrefs

Cf. A090989-A090995.

Partial sums of pairwise sums of A065455.

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

Sequence in context: A032690 A188557 A356966 * A259215 A215333 A034780

Adjacent sequences: A090989 A090990 A090991 * A090993 A090994 A090995

Keyword

nonn,easy

Author

Branko Malesevic, Feb 29 2004

Extensions

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

More terms from Joseph Myers, Dec 23 2008

Status

approved