A154355 a(n) = 25*n^2 - 36*n + 13.

13, 2, 41, 130, 269, 458, 697, 986, 1325, 1714, 2153, 2642, 3181, 3770, 4409, 5098, 5837, 6626, 7465, 8354, 9293, 10282, 11321, 12410, 13549, 14738, 15977, 17266, 18605, 19994, 21433, 22922, 24461, 26050, 27689, 29378

Offset: 0

Author

Vincenzo Librandi, Jan 07 2009

Keywords

• nonn
• easy

Comments

The identity (1250*n^2 - 1800*n + 649)^2 - (25*n^2 - 36*n + 13)*(250*n - 180)^2 = 1 can be written as A154358(n)^2 - a(n)*A154360(n)^2 = 1. See also the third comment in A154357.

Numbers of the form (3n-2)^2 + (4n-3)^2. - Bruno Berselli, Dec 12 2011

From Klaus Purath, May 06 2025: (Start)

25*a(n)-1 is a square, and a(n) is the sum of two squares (see FORMULA). There are no squares in this sequence. The odd prime factors of these terms are always of the form 4*k + 1.

All a(n) = D satisfy the Pell equation (k*x)^2 - D*(5*y)^2 = -1 for any integer n where a(1-n) = A154357(n). The values for k and the solutions x, y can be calculated using the following algorithm: k = sqrt(D*5^2 - 1), x(0) = 1, x(1) = 4*D*5^2 - 1, y(0) = 1, y(1) = 4*D*5^2 - 3. The two recurrences are of the form (4*D*5^2 - 2, -1).

It follows from the above that the terms of this sequence and of A154357 belong to A031396. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A154354, A154358, A154359, A154360, A154361, A202141.

Essentially a duplicate of A007533.

Programs

Magma

[25*n^2-36*n+13: n in [0..40]]; // Bruno Berselli, Sep 15 2016

Mathematica

Table[25n^2-36n+13,{n,0,40}]  (* Harvey P. Dale, Apr 02 2011 *)
LinearRecurrence[{3, -3, 1}, {13, 2, 41}, 50] (* Vincenzo Librandi, Feb 21 2012 *)

PARI

for(n=0, 40, print1(25*n^2 - 36*n + 13", ")); \\ Vincenzo Librandi, Feb 21 2012

Formula

a(n) = A007533(n-1), n>0. - R. J. Mathar, Jan 14 2009

G.f.: (13 - 37*x + 74*x^2) / (1-x)^3. - R. J. Mathar, Jan 05 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 21 2012

E.g.f.: (13 - 11*x + 25*x^2) * exp(x). - G. C. Greubel, Sep 14 2016

From Klaus Purath, May 06 2025: (Start)

a(n) = (3*n-2)^2 + (4*n-3)^2.

25*a(n) - 1 = (25*n - 18)^2. (End)

Extensions

Offset corrected from R. J. Mathar, Jan 05 2011

First comment rewritten by Bruno Berselli, Dec 12 2011