OFFSET
1,2
COMMENTS
Generating polynomial is Schur's polynomial of degree 3. Schur's polynomials n degree are n-th first term of series expansion of e^x function. All polynomials are non-reducible and belonging to the An alternating Galois transitive group if n is divisible by 4 or to Sn symmetric Galois Group in other case (proof Schur, 1930).
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=4, a(n-2)=-coeff(charpoly(A,x),x^(n-3)). - Milan Janjic, Jan 27 2010
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
From Colin Barker, May 15 2016: (Start)
a(n) = (9*n^3-18*n^2+15*n-4)/2.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x*(1+2*x)*(1+7*x+x^2) / (1-x)^4.
(End)
E.g.f.: 2 + (9*x^3 + 9*x^2 + 6*x - 4)*exp(x)/2. - G. C. Greubel, Apr 29 2018
MATHEMATICA
a = {}; Do[If[IntegerQ[1 + x + x^2/2 + x^3/6], AppendTo[a, 1 + x + x^2/2 + x^3/6]], {x, 1, 300}]; a
Select[Table[x^3/6 + x^2/2 + x + 1, {x, 0, 200}], IntegerQ] (* Harvey P. Dale, Jan 06 2011 *)
PROG
(PARI) Vec(x*(1+2*x)*(1+7*x+x^2)/(1-x)^4 + O(x^50)) \\ Colin Barker, May 15 2016
(Magma) [(9*n^3-18*n^2+15*n-4)/2: n in [1..30]]; // G. C. Greubel, Apr 29 2018
(GAP) Filtered(List([0..150], x->(x^3)/6+(x^2)/2+x+1), IsInt); # Muniru A Asiru, Apr 30 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Feb 04 2007
EXTENSIONS
a(1) = 1 added by Harvey P. Dale, Jan 06 2011
STATUS
approved