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%I #28 Sep 08 2022 08:45:29
%S 1,13,61,172,373,691,1153,1786,2617,3673,4981,6568,8461,10687,13273,
%T 16246,19633,23461,27757,32548,37861,43723,50161,57202,64873,73201,
%U 82213,91936,102397,113623,125641,138478,152161,166717,182173,198556,215893,234211
%N Integers of the form (x^3)/6 + (x^2)/2 + x + 1.
%C 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).
%C 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
%H Colin Barker, <a href="/A127876/b127876.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F From _Colin Barker_, May 15 2016: (Start)
%F a(n) = (9*n^3-18*n^2+15*n-4)/2.
%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
%F G.f.: x*(1+2*x)*(1+7*x+x^2) / (1-x)^4.
%F (End)
%F E.g.f.: 2 + (9*x^3 + 9*x^2 + 6*x - 4)*exp(x)/2. - _G. C. Greubel_, Apr 29 2018
%t 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
%t Select[Table[x^3/6 + x^2/2 + x + 1, {x, 0, 200}], IntegerQ] (* _Harvey P. Dale_, Jan 06 2011 *)
%o (PARI) Vec(x*(1+2*x)*(1+7*x+x^2)/(1-x)^4 + O(x^50)) \\ _Colin Barker_, May 15 2016
%o (Magma) [(9*n^3-18*n^2+15*n-4)/2: n in [1..30]]; // _G. C. Greubel_, Apr 29 2018
%o (GAP) Filtered(List([0..150],x->(x^3)/6+(x^2)/2+x+1),IsInt); # _Muniru A Asiru_, Apr 30 2018
%Y Cf. A127873, A127874, A127875.
%K nonn,easy
%O 1,2
%A _Artur Jasinski_, Feb 04 2007
%E a(1) = 1 added by _Harvey P. Dale_, Jan 06 2011
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