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a(n) = 5*n^2 - 4*n + 1.
7

%I #32 Dec 03 2023 18:19:43

%S 1,2,13,34,65,106,157,218,289,370,461,562,673,794,925,1066,1217,1378,

%T 1549,1730,1921,2122,2333,2554,2785,3026,3277,3538,3809,4090,4381,

%U 4682,4993,5314,5645,5986,6337,6698,7069,7450,7841,8242,8653,9074

%N a(n) = 5*n^2 - 4*n + 1.

%C 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 Vincenzo Librandi, <a href="/A190816/b190816.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Harvey P. Dale_, May 24 2011: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=2, a(2)=13.

%F G.f.: (1 - x + 10*x^2)/(1-x)^3. (End)

%F E.g.f.: (1 + x + 5*x^2)*exp(x). - _G. C. Greubel_, Dec 03 2023

%t Table[5*n^2 - 4*n + 1, {n, 0, 100}]

%t 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 (Magma) [5*n^2 - 4*n + 1: n in [0..50]]; // _Vincenzo Librandi_, Jun 19 2011

%o (PARI) a(n)=5*n^2-4*n+1 \\ _Charles R Greathouse IV_, Oct 16 2015

%o (SageMath) [5*n^2-4*n+1 for n in range(41)] # _G. C. Greubel_, Dec 03 2023

%Y Short sides (a) A045944(n-1), long sides (b) A002939(n).

%Y Cf. A017281 (first differences), A051624 (a(n)-1), A202141.

%Y 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 nonn,easy

%O 0,2

%A _Vladimir Joseph Stephan Orlovsky_, May 20 2011

%E Edited by _Franklin T. Adams-Watters_, May 20 2011