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
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