A158485 a(n) = 14*n^2 - 1.

13, 55, 125, 223, 349, 503, 685, 895, 1133, 1399, 1693, 2015, 2365, 2743, 3149, 3583, 4045, 4535, 5053, 5599, 6173, 6775, 7405, 8063, 8749, 9463, 10205, 10975, 11773, 12599, 13453, 14335, 15245, 16183, 17149, 18143, 19165, 20215, 21293, 22399

Offset

1,1

Comments

The identity (14*n^2-1)^2-(49*n^2-7)*(2*n)^2=1 can be written as a(n)^2-A158484(n)*A005843(n)^2=1.

Sequence found by reading the line from 13, in the direction 13, 55,..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 13 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

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

G.f: x*(-13-16*x+x^2)/(x-1)^3.

From Amiram Eldar, Feb 04 2021: (Start)

Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(14))*cot(Pi/sqrt(14)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(14))*csc(Pi/sqrt(14)) - 1)/2.

Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(14))*csc(Pi/sqrt(14)).

Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(14))*sin(Pi/sqrt(7))/sqrt(2). (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {13, 55, 125}, 50]

Prog

(Magma) I:=[13, 55, 125]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
(PARI) a(n) = 14*n^2-1

Crossrefs

Cf. A005843, A158484.

Sequence in context: A027000 A198160 A029531 * A274973 A005902 A051798

Adjacent sequences: A158482 A158483 A158484 * A158486 A158487 A158488

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 20 2009

Status

approved