A125202 a(n) = 4*n^2 - 6*n + 1.

-1, 5, 19, 41, 71, 109, 155, 209, 271, 341, 419, 505, 599, 701, 811, 929, 1055, 1189, 1331, 1481, 1639, 1805, 1979, 2161, 2351, 2549, 2755, 2969, 3191, 3421, 3659, 3905, 4159, 4421, 4691, 4969, 5255, 5549, 5851, 6161, 6479, 6805, 7139, 7481, 7831, 8189, 8555, 8929, 9311, 9701, 10099, 10505, 10919, 11341

Offset: 1

Author

Reinhard Zumkeller, Nov 24 2006

Keywords

• sign
• easy

Links

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

Crossrefs

Cf. A125199.

Cf. A003415, A017089.

Cf. A002939, A016789, A017041, A017485, A028387.

Cf. A002943, A028387.

Programs

Magma

[4*n^2 - 6*n + 1: n in [1..60]]; // Vincenzo Librandi, Jul 11 2011

Mathematica

f[a_]:=4*a^2-6*a+1; lst={};Do[AppendTo[lst,f[n]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 14 2009 *)

PARI

a(n)=4*n^2-6*n+1 \\ Charles R Greathouse IV, Sep 28 2015

Formula

a(n) = A125199(n,n-1) for n>1.

A003415(a(n)) = A017089(n-1).

From Arkadiusz Wesolowski, Dec 25 2011: (Start)

a(1) = -1, a(n) = a(n-1) + 8*n - 10.

a(n) = 2*a(n-1) - a(n-2) + 8 with a(1) = -1 and a(2) = 5.

G.f.: (1 - 4*x + 11*x^2)/(1 - x)^3. (End)

a(n) = A002943(n-1) - 1. - Arkadiusz Wesolowski, Feb 15 2012

a(n) = A028387(2n-3), with A028387(-1) = -1. - Vincenzo Librandi, Oct 10 2013

E.g.f.: exp(x)*(1 - 2*x + 4*x^2). - Stefano Spezia, Oct 10 2022

Sum_{n>=1} 1/a(n) = sqrt(5)/10*(psi(1/4+sqrt(5)/4) - psi(1/4-sqrt(5)/4)) = -0.656213833... - R. J. Mathar, Apr 22 2024