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 A190816 #38 May 17 2025 05:15:41
%S A190816 1,2,13,34,65,106,157,218,289,370,461,562,673,794,925,1066,1217,1378,
%T A190816 1549,1730,1921,2122,2333,2554,2785,3026,3277,3538,3809,4090,4381,
%U A190816 4682,4993,5314,5645,5986,6337,6698,7069,7450,7841,8242,8653,9074
%N A190816 a(n) = 5*n^2 - 4*n + 1.
%C A190816 For n >= 2, hypotenuses of primitive Pythagorean triangles with m = 2*n-1, where the sides of the triangle are a = m^2 - n^2, b = 2*n*m, c = m^2 + n^2; this sequence is the c values, short sides (a) are A045944(n-1), and long sides (b) are A002939(n).
%H A190816 Vincenzo Librandi, <a href="/A190816/b190816.txt">Table of n, a(n) for n = 0..2000</a>
%H A190816 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A190816 From _Harvey P. Dale_, May 24 2011: (Start)
%F A190816 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=13.
%F A190816 G.f.: (1 - x + 10*x^2)/(1-x)^3. (End)
%F A190816 E.g.f.: (1 + x + 5*x^2)*exp(x). - _G. C. Greubel_, Dec 03 2023
%t A190816 Table[5*n^2 - 4*n + 1, {n, 0, 100}]
%t A190816 LinearRecurrence[{3,-3,1},{1,2,13},100] (* or *) CoefficientList[ Series[ (-10 x^2+x-1)/(x-1)^3,{x,0,100}],x] (* _Harvey P. Dale_, May 24 2011 *)
%o A190816 (Magma) [5*n^2 - 4*n + 1: n in [0..50]]; // _Vincenzo Librandi_, Jun 19 2011
%o A190816 (PARI) a(n)=5*n^2-4*n+1 \\ _Charles R Greathouse IV_, Oct 16 2015
%o A190816 (SageMath) [5*n^2-4*n+1 for n in range(41)] # _G. C. Greubel_, Dec 03 2023
%Y A190816 Short sides (a) A045944(n-1), long sides (b) A002939(n).
%Y A190816 Cf. A017281 (first differences), A051624 (a(n)-1), A202141.
%Y A190816 Sequences of the form m*n^2 - 4*n + 1: -A131098 (m=0), A028872 (m=1), A056220 (m=2), A045944 (m=3), A016754 (m=4), this sequence (m=5), A126587 (m=6), A339623 (m=7), A080856 (m=8).
%K A190816 nonn,easy
%O A190816 0,2
%A A190816 _Vladimir Joseph Stephan Orlovsky_, May 20 2011
%E A190816 Edited by _Franklin T. Adams-Watters_, May 20 2011