This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A080856 #44 Jul 24 2025 17:58:10
%S A080856 1,5,25,61,113,181,265,365,481,613,761,925,1105,1301,1513,1741,1985,
%T A080856 2245,2521,2813,3121,3445,3785,4141,4513,4901,5305,5725,6161,6613,
%U A080856 7081,7565,8065,8581,9113,9661,10225,10805,11401,12013,12641,13285,13945,14621
%N A080856 a(n) = 8*n^2 - 4*n + 1.
%C A080856 The old definition of this sequence was "Generalized polygonal numbers".
%C A080856 Row T(4,n) of A080853.
%C A080856 {a(k): 0 <= k < 3} = divisors of 25. - _Reinhard Zumkeller_, Jun 17 2009
%C A080856 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
%C A080856 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
%C A080856 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
%H A080856 G. C. Greubel, <a href="/A080856/b080856.txt">Table of n, a(n) for n = 0..5000</a>
%H A080856 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H A080856 Reinhard Zumkeller, <a href="/A161700/a161700.txt">Enumerations of Divisors</a>.
%H A080856 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A080856 G.f.: (1+2*x+13*x^2)/(1-x)^3.
%F A080856 a(n) = A060820(n), n>0. - _R. J. Mathar_, Sep 18 2008
%F A080856 a(n) = C(n,0) + 4*C(n,1) + 16*C(n,2). - _Reinhard Zumkeller_, Jun 17 2009
%F A080856 a(n) = 16*n+a(n-1)-12 with n>0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010
%F A080856 E.g.f.: (8*x^2 + 4*x + 1)*exp(x). - _G. C. Greubel_, Jun 16 2017
%p A080856 A080856:=n->8*n^2 - 4*n + 1: seq(A080856(n), n=0..100); # _Wesley Ivan Hurt_, Jul 16 2017
%t A080856 LinearRecurrence[{3, -3, 1}, {1, 5, 25}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 17 2012 *)
%o A080856 (PARI) a(n)=8*n^2-4*n+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A080856 Cf. A005408, A000124, A016813, A049061, A080853, A086514, A000125, A058331, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A161711, A161712, A161713, A161715, A006261.
%Y A080856 A060820 is another version (but the present sequence is the main entry).
%Y A080856 A row of the array in A386478.
%K A080856 nonn,easy
%O A080856 0,2
%A A080856 _Paul Barry_, Feb 23 2003
%E A080856 Definition replaced with the closed form by _Bruno Berselli_, Jan 16 2013