A158447 a(n) = 10*n^2 - 1.

9, 39, 89, 159, 249, 359, 489, 639, 809, 999, 1209, 1439, 1689, 1959, 2249, 2559, 2889, 3239, 3609, 3999, 4409, 4839, 5289, 5759, 6249, 6759, 7289, 7839, 8409, 8999, 9609, 10239, 10889, 11559, 12249, 12959, 13689, 14439, 15209, 15999, 16809, 17639

Offset: 1

Author

Vincenzo Librandi, Mar 19 2009

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 9, in the direction 9, 39, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Vincenzo Librandi, X^2-AY^2=1.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005843, A033583, A085787, A158446.

Programs

Magma

I:=[9, 39, 89]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

Maple

A158447:=n->10*n^2-1: seq(A158447(n), n=1..100); # Wesley Ivan Hurt, Apr 26 2017

Mathematica

Table[10n^2-1,{n,50}]
LinearRecurrence[{3,-3,1},{9,39,89},50] (* Harvey P. Dale, Dec 08 2017 *)

PARI

a(n) = 10*n^2 - 1.

Formula

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

G.f: x*(9 + 12*x - x^2)/(1 - x)^3.

a(n) = A033583(n) - 1. - Omar E. Pol, Jul 18 2012

From Amiram Eldar, Feb 04 2021: (Start)

Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(10))*cot(Pi/sqrt(10)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(10))*csc(Pi/sqrt(10)) - 1)/2.

Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(10))*csc(Pi/sqrt(10)).

Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(10))*sin(Pi/sqrt(5))/sqrt(2). (End)

E.g.f.: exp(x)*(10*x^2 + 10*x - 1) + 1. - Stefano Spezia, Aug 25 2022