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A080856
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8*n^2 - 4*n + 1.
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23
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1, 5, 25, 61, 113, 181, 265, 365, 481, 613, 761, 925, 1105, 1301, 1513, 1741, 1985, 2245, 2521, 2813, 3121, 3445, 3785, 4141, 4513, 4901, 5305, 5725, 6161, 6613, 7081, 7565, 8065, 8581, 9113, 9661, 10225, 10805, 11401, 12013, 12641, 13285, 13945, 14621
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OFFSET
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0,2
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COMMENTS
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The old definition of this sequence was "Generalized polygonal numbers".
Row T(4,n) of A080853.
{a(k): 0 <= k < 3} = divisors of 25. [Reinhard Zumkeller, Jun 17 2009]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=4, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=3, a(n-1)= coeff(charpoly(A,x),x^(n-2)). [Milan Janjic, Jan 27 2010]
Also sequence found by reading the segment (1, 5) together with the line from 5, in the direction 5, 25,..., in the square spiral whose vertices are the generalized hexagonal numbers A000217. - Omar E. Pol, Nov 05 2012
For n > 0: A049061(a(n)) = 0, when the triangle of "signed Eulerian numbers" in A049061 is seen as flattened sequence. - Reinhard Zumkeller, Jan 31 2013
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LINKS
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Table of n, a(n) for n=0..43.
R. Zumkeller, Enumerations of Divisors.
Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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G.f.: (1+2*x+13*x^2)/(1-x)^3.
a(n) = A060820(n), n>0. [R. J. Mathar, Sep 18 2008]
a(n) = C(n,0) + 4*C(n,1) + 16*C(n,2). [Reinhard Zumkeller, Jun 17 2009]
a(n) = 16*n+a(n-1)-12 with n>0, a(0)=1. [Vincenzo Librandi, Aug 08 2010]
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1, 5, 25}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
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CROSSREFS
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Cf. A005408, A000124, A016813, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A006261.
Sequence in context: A146649 A146412 A152734 * A060820 A182211 A146404
Adjacent sequences: A080853 A080854 A080855 * A080857 A080858 A080859
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KEYWORD
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nonn,easy
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AUTHOR
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Paul Barry, Feb 23 2003
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EXTENSIONS
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Definition replaced with the closed form by Bruno Berselli, Jan 16 2013
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STATUS
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approved
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