A158128 a(n) = 100*n + 1.

101, 201, 301, 401, 501, 601, 701, 801, 901, 1001, 1101, 1201, 1301, 1401, 1501, 1601, 1701, 1801, 1901, 2001, 2101, 2201, 2301, 2401, 2501, 2601, 2701, 2801, 2901, 3001, 3101, 3201, 3301, 3401, 3501, 3601, 3701, 3801, 3901, 4001, 4101, 4201, 4301, 4401

Offset: 1

Author

Vincenzo Librandi, Mar 13 2009

Keywords

• nonn
• easy

Comments

The identity (100*n+1)^2-(100*n^2+2*n)*10^2 = 1 can be written as a(n)^2-A158127(n)*10^2 = 1. - Vincenzo Librandi, Feb 11 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations. (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(10^2*t+2)).
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A158127.

Programs

Magma

I:=[101, 201]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 11 2012

Mathematica

LinearRecurrence[{2, -1}, {101, 201}, 50] (* Vincenzo Librandi, Feb 11 2012 *)
100*Range[50]+1 (* Harvey P. Dale, Apr 12 2024 *)

PARI

for(n=1, 40, print1(100*n + 1", ")); \\ Vincenzo Librandi, Feb 11 2012

Formula

From Vincenzo Librandi, Feb 11 2012: (Start)

G.f.: x*(101-x)/(1-x)^2.

a(n) = 2*a(n-1)-a(n-2). (End)

E.g.f.: exp(x)*(100*x + 1) - 1. - Stefano Spezia, Jan 29 2025