A190816 a(n) = 5*n^2 - 4*n + 1.

1, 2, 13, 34, 65, 106, 157, 218, 289, 370, 461, 562, 673, 794, 925, 1066, 1217, 1378, 1549, 1730, 1921, 2122, 2333, 2554, 2785, 3026, 3277, 3538, 3809, 4090, 4381, 4682, 4993, 5314, 5645, 5986, 6337, 6698, 7069, 7450, 7841, 8242, 8653, 9074

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, May 20 2011

Keywords

• nonn
• easy

Comments

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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

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

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).

Programs

Magma

[5*n^2 - 4*n + 1: n in [0..50]]; // Vincenzo Librandi, Jun 19 2011

Mathematica

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

PARI

a(n)=5*n^2-4*n+1 \\ Charles R Greathouse IV, Oct 16 2015

SageMath

[5*n^2-4*n+1 for n in range(41)] # G. C. Greubel, Dec 03 2023

Formula

From Harvey P. Dale, May 24 2011: (Start)

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

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

E.g.f.: (1 + x + 5*x^2)*exp(x). - G. C. Greubel, Dec 03 2023

Extensions

Edited by Franklin T. Adams-Watters, May 20 2011