A164582 a(n) = 5*a(n - 1) - a(n - 2), with n>2, a(1)=2, a(2)=3.

2, 3, 13, 62, 297, 1423, 6818, 32667, 156517, 749918, 3593073, 17215447, 82484162, 395205363, 1893542653, 9072507902, 43468996857, 208272476383, 997893385058, 4781194448907, 22908078859477, 109759199848478, 525887920382913, 2519680402066087

Offset: 1

Author

Vincenzo Librandi, Aug 17 2009

Keywords

• nonn
• easy

Comments

From Klaus Purath, Aug 18 2024: (Start)

For any two consecutive terms (x,y), x^2 - 5xy + y^2 = -17 = A127147(6) always applies. In general, the following applies to all recurrences (t) with constant coefficients (5,-1) and t(0) = 2 and two consecutive terms (x,y): x^2 - 5xy + y^2 = A127147(t(1)+3) for any integer t(1). This includes and interprets the Feb 08 2014 comment on A003501 by Colin Barker.

By analogy to this, for three consecutive terms (x,y,z) of any recurrence (t) of the form (5,-1) with t(0) = 2: y^2 - xz = A127147(t(1)+3).

a(n) = t(n) - t(n-1) = (t(n+1) - t(n-2))/6, where (t) is any third order recurrence with constant coefficients (6,-6,1) and initial values t(0) = x, t(1) = x + 2, t(2) = x + 5 for any integer x.

a(n) = t(n-1) + t(n) = (t(n-2) + t(n+1))/4, where (t) is any third order recurrence with constant coefficients (4,4,-1) and initial values t(0) = x, t(1) = 2 - x, t(2) = x + 1 for any integer x. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-1).

Programs

Magma

[n le 2 select n+1 else 5*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 12 2013

Mathematica

CoefficientList[Series[(2 - 7 x) / (1 - 5 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 12 2013 *)
LinearRecurrence[{5,-1},{2,3},30] (* Harvey P. Dale, Apr 06 2016 *)

PARI

Vec(x*(2 - 7*x) / (1 - 5*x + x^2) + O(x^30)) \\ Colin Barker, Nov 08 2017

Formula

a(n) = 5*a(n-1) - a(n-2) = 2*A004254(n) - 7*A004254(n-1).

G.f.: x*(2-7*x) / (1-5*x+x^2).

a(n) = (2^(-1-n)*((5+sqrt(21))^n*(-31+7*sqrt(21)) + (5-sqrt(21))^n*(31+7*sqrt(21)))) / sqrt(21). - Colin Barker, Nov 08 2017

a(n) = (a(n-1)^2 + 17)/a(n-2). - Klaus Purath, Aug 30 2020

Extensions

Extended by R. J. Mathar, Aug 19 2009