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A090992
Number of meaningful differential operations of the n-th order on the space R^7.
6
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
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
Partial sums of pairwise sums of A065455.
Sequence in context: A032690 A188557 A356966 * A259215 A215333 A034780
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