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A158480 12n^2 + 1. 2
13, 49, 109, 193, 301, 433, 589, 769, 973, 1201, 1453, 1729, 2029, 2353, 2701, 3073, 3469, 3889, 4333, 4801, 5293, 5809, 6349, 6913, 7501, 8113, 8749, 9409, 10093, 10801, 11533, 12289, 13069, 13873, 14701, 15553, 16429, 17329, 18253, 19201 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The identity (12*n^2+1)^2 - (36*n^2+6)*(2*n)^2 = 1 can be written as a(n)^2 - A158479(n)*A005843(n)^2 = 1.

Sequence found by reading the line from 13, in the direction 13, 49, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - 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).

FORMULA

a(n) = A010014(n)/2. - Vladimir Joseph Stephan Orlovsky, May 18 2009

G.f: x*(13+10*x+x^2)/(1-x)^3.

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

a(n) = 1 + A135453(n). - Omar E. Pol, Jul 18 2012

EXAMPLE

For n=1, a(1)=13; n=2, a(2)=49; n=3, a(3)=109.

MATHEMATICA

a=1; lst={}; Do[b=n^3-a; AppendTo[lst, b/2]; a+=b, {n, 3, 6!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, May 18 2009 *)

LinearRecurrence[{3, -3, 1}, {13, 49, 109}, 40]

PROG

(MAGMA) I:=[13, 49, 109]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];

(PARI) a(n)=12*n^2+1

CROSSREFS

Cf. A005843, A158479.

Sequence in context: A080171 A044115 A044496 * A009951 A274974 A251142

Adjacent sequences:  A158477 A158478 A158479 * A158481 A158482 A158483

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Mar 20 2009

STATUS

approved

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Last modified June 25 11:31 EDT 2017. Contains 288709 sequences.