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A090991
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Number of meaningful differential operations of the n-th order on the space R^6.
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8
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6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972
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OFFSET
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1,1
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COMMENTS
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Apparently a(n) = A054886(n+2) for n=1..1000. - Georg Fischer, Oct 06 2018
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LINKS
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Indranil Ghosh, Table of n, a(n) for n = 1..4772
Tanya Khovanova, Recursive Sequences
Branko J. 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 J. Malesevic, Some combinatorial aspects of differential operation compositions on space R^n, arXiv:0704.0750 [math.DG], 2007.
Index entries for linear recurrences with constant coefficients, signature (1,1).
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FORMULA
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a(k+4) = 3*a(k+2) - a(k).
a(k) = 2*Fibonacci(k+3).
From Philippe Deléham, Nov 19 2008: (Start)
a(n) = a(n-1) + a(n-2), n>2, where a(1)=6, a(2)=10.
G.f.: 2*x*(3+2*x)/(1-x-x^2). (End)
E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 4. - Stefano Spezia, Apr 18 2022
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MAPLE
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NUM := proc(k :: integer) local i, j, n, Fun, Identity, v, A; n := 6; # <- 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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CoefficientList[Series[2*(3+2z)/(1-z-z^2), {z, 0, 40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
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PROG
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(GAP) a:=[6, 10];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]; od; a; # Muniru A Asiru, Oct 06 2018
(PARI) my(x='x+O('x^40)); Vec(2*x*(3+2*x)/(1-x-x^2)) \\ G. C. Greubel, Feb 02 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( 2*x*(3+2*x)/(1-x-x^2) )); // G. C. Greubel, Feb 02 2019
(Sage) (2*(3+2*x)/(1-x-x^2)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 02 2019
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CROSSREFS
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Cf. A054886, A055389, A068922, A078642, A090989-A090995.
Essentially the same as A006355.
Sequence in context: A302748 A020741 A274287 * A019533 A053301 A262542
Adjacent sequences: A090988 A090989 A090990 * A090992 A090993 A090994
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KEYWORD
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nonn,easy
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AUTHOR
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Branko Malesevic, Feb 29 2004
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STATUS
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approved
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