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A090994
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Number of meaningful differential operations of the n-th order on the space R^9.
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8
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9, 17, 32, 61, 116, 222, 424, 813, 1556, 2986, 5721, 10982, 21053, 40416, 77505, 148785, 285380, 547810, 1050876, 2017126, 3869845, 7427671, 14250855, 27351502, 52479500, 100719775, 193258375, 370895324, 711682501, 1365808847, 2620797529
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OFFSET
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1,1
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COMMENTS
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Also number of meaningful compositions of the n-th order of the differential operations and Gateaux directional derivative on the space R^8. - Branko Malesevic and Ivana Jovovic (ivana121(AT)EUnet.yu), Jun 21 2007
Also (starting 5,9,...) the number of zig-zag paths from top to bottom of a rectangle of width 10, whose color is that of the top right corner. [From Joseph Myers, Dec 23 2008]
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LINKS
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FORMULA
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a(k+5) = a(k+4) + 4*a(k+3) - 3*a(k+2) - 3*a(k+1) + a(k).
G.f.: x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; corrected by R. J. Mathar, Sep 16 2009
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MAPLE
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NUM := proc(k :: integer) local i, j, n, Fun, Identity, v, A; n := 9; # <- 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
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LinearRecurrence[{1, 4, -3, -3, 1}, {9, 17, 32, 61, 116}, 31] (* Jean-François Alcover, Nov 20 2017 *)
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PROG
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(PARI) my(x='x+O('x^40)); Vec(x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2 +3*x^3+3*x^4-x^5)) \\ G. C. Greubel, Feb 02 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5) )); // G. C. Greubel, Feb 02 2019
(Sage) a=(x*(9+8*x-21*x^2-12*x^3+5*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 02 2019
(GAP) a:=[9, 17, 32, 61, 116];; for n in [6..40] do a[n]:=a[n-1]+4*a[n-2] - 3*a[n-3]-3*a[n-4]+a[n-5]; od; a; # G. C. Greubel, Feb 02 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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