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A127876
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Integers of the form (x^3)/6 + (x^2)/2 + x + 1.
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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)). [From Milan R. Janjic (agnus(AT)blic.net), Jan 27 2010]
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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 (* From Artur Jasinski *)
Select[Table[x^3/6 + x^2/2 + x + 1, {x, 0, 200}], IntegerQ] (* From Harvey P. Dale, Jan 06 2011 *)
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CROSSREFS
| Cf. A127873, A127874, A127875.
Sequence in context: A119151 A081589 A139880 * A047673 A141725 A147185
Adjacent sequences: A127873 A127874 A127875 * A127877 A127878 A127879
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KEYWORD
| nonn
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AUTHOR
| Artur Jasinski (grafix(AT)csl.pl), Feb 04 2007
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EXTENSIONS
| Added a(1) = 1, Harvey P. Dale, Jan 06 2011
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