A158491 a(n) = 20*n^2 - 1.

19, 79, 179, 319, 499, 719, 979, 1279, 1619, 1999, 2419, 2879, 3379, 3919, 4499, 5119, 5779, 6479, 7219, 7999, 8819, 9679, 10579, 11519, 12499, 13519, 14579, 15679, 16819, 17999, 19219, 20479, 21779, 23119, 24499, 25919, 27379, 28879, 30419, 31999, 33619, 35279

Offset: 1

Author

Vincenzo Librandi, Mar 20 2009

Keywords

• nonn
• easy

Comments

The identity (20*n^2 - 1)^2 - (100*n^2 - 10)*(2*n)^2 = 1 can be written as a(n)^2 - A158490(n)*A005843(n)^2 = 1.

Sequence found by reading the line from 19, in the direction 19, 79, ... in the square spiral whose vertices are the generalized dodecagonal numbers A195162. - Omar E. Pol, Nov 05 2012

Links

• 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).

Crossrefs

Cf. A005843, A134538, A158490, A195162.

Programs

Magma

I:=[19, 79, 179]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

Mathematica

LinearRecurrence[{3,-3,1},{19,79,179},50]
20*Range[40]^2-1 (* Harvey P. Dale, Aug 24 2021 *)

PARI

a(n)=20*n^2-1 \\ Charles R Greathouse IV, Dec 23 2011

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f: x*(-19-22*x+x^2)/(x-1)^3.

From Amiram Eldar, Mar 06 2023: (Start)

Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(5)))*Pi/(2*sqrt(5)) - 1)/2. (End)

From Elmo R. Oliveira, Jan 25 2025: (Start)

E.g.f.: exp(x)*(20*x^2 + 20*x - 1) + 1.

a(n) = A134538(2*n). (End)