A127876 Integers of the form (x^3)/6 + (x^2)/2 + x + 1.

1, 13, 61, 172, 373, 691, 1153, 1786, 2617, 3673, 4981, 6568, 8461, 10687, 13273, 16246, 19633, 23461, 27757, 32548, 37861, 43723, 50161, 57202, 64873, 73201, 82213, 91936, 102397, 113623, 125641, 138478, 152161, 166717, 182173, 198556, 215893, 234211

Offset: 1

Author

Artur Jasinski, Feb 04 2007

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A127873, A127874, A127875.

Programs

GAP

Filtered(List([0..150],x->(x^3)/6+(x^2)/2+x+1),IsInt); # Muniru A Asiru, Apr 30 2018

Magma

[(9*n^3-18*n^2+15*n-4)/2: n in [1..30]]; // 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 *)

PARI

Vec(x*(1+2*x)*(1+7*x+x^2)/(1-x)^4 + O(x^50)) \\ Colin Barker, May 15 2016

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

Extensions

a(1) = 1 added by Harvey P. Dale, Jan 06 2011