A158602 -id:A158602 - cp's OEIS frontend

A158601 a(n) = 400*n^2 + 20.

20, 420, 1620, 3620, 6420, 10020, 14420, 19620, 25620, 32420, 40020, 48420, 57620, 67620, 78420, 90020, 102420, 115620, 129620, 144420, 160020, 176420, 193620, 211620, 230420, 250020, 270420, 291620, 313620, 336420, 360020, 384420, 409620, 435620, 462420, 490020

Offset: 0

Vincenzo Librandi, Mar 22 2009

The identity (40*n^2 + 1)^2 - (400*n^2 + 20)*(2*n)^2 = 1 can be written as A158602(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158493, A158602.

I:=[20, 420, 1620]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 16 2012

400 Range[0,40]^2+20  (* Harvey P. Dale, Feb 05 2011 *)
LinearRecurrence[{3, -3, 1}, {20, 420, 1620}, 50] (* Vincenzo Librandi, Feb 16 2012 *)

for(n=0, 40, print1(400*n^2 + 20", ")); \\ Vincenzo Librandi, Feb 16 2012

G.f.: -20*(1 + 18*x + 21*x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 16 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) + 1)/40.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) + 1)/40. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 20*exp(x)*(1 + 20*x + 20*x^2).
a(n) = 20*A158493(n). (End)

Comment rewritten, formula replaced by R. J. Mathar, Oct 28 2009

A158775 a(n) = 1600*n^2 + 40.

Original entry on oeis.org

1640, 6440, 14440, 25640, 40040, 57640, 78440, 102440, 129640, 160040, 193640, 230440, 270440, 313640, 360040, 409640, 462440, 518440, 577640, 640040, 705640, 774440, 846440, 921640, 1000040, 1081640, 1166440, 1254440, 1345640, 1440040, 1537640, 1638440, 1742440

Offset: 1

Vincenzo Librandi, Mar 26 2009

The identity (80*n^2 + 1)^2 - (1600*n^2 + 40)*(2*n)^2 = 1 can be written as A158776(n)^2 - a(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A158602, A158776.

I:=[1640, 6440, 14440]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

LinearRecurrence[{3, -3, 1}, {1640, 6440, 14440}, 50] (* Vincenzo Librandi, Feb 20 2012 *)
1600*Range[30]^2+ 40 (* Harvey P. Dale, Jun 02 2020 *)

for(n=1, 40, print1(1600*n^2 + 40", ")); \\ Vincenzo Librandi, Feb 20 2012

From R. J. Mathar, Jul 26 2009: (Start)
G.f.: -40*x*(41 + 38*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 24 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)) - 1)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/(2*sqrt(10)))*Pi/(2*sqrt(10)))/80. (End)
From Elmo R. Oliveira, Jan 26 2025: (Start)
E.g.f.: 40*(exp(x)*(40*x^2 + 40*x + 1) - 1).
a(n) = 40*A158602(n). (End)

Edited by R. J. Mathar, Jul 26 2009

cp's OEIS Frontend

A158601 a(n) = 400*n^2 + 20.

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