A147525 Numbers X such that there exists Y in N : X^2 = 309*Y^2 + 103.

1818362, 233487592671260018, 29981079637522561422843699458, 3849734052023190225799987829591837303402, 494326837141597862882377389715613614004616427568522, 63474260459627156072628000315760330760620752336069144700433378

Offset

1,1

Comments

Obtained with the fundamental unit 64202725495 + 3652365444*sqrt(309) of Q(sqrt(309)).

Links

G. C. Greubel, Table of n, a(n) for n = 1..50

Index entries for linear recurrences with constant coefficients, signature (128405450990,-1).

Formula

a(n+2) = 128405450990*a(n+1) - a(n).

G.f.: 1818362*x*(1-x)/(1 - 128405450990*x + x^2). - Colin Barker, Oct 21 2014

Example

a(1)=1818362 because the first relation is : 1818362^2=309*103443^2+103.

Maple

seq(coeff(series(1818362*x*(1-x)/(1-128405450990*x+x^2), x, n+1), x, n), n = 1..10); # G. C. Greubel, Jan 10 2020

Mathematica

LinearRecurrence[{128405450990, -1}, {1818362, 233487592671260018}, 10] (* G. C. Greubel, Jan 10 2020 *)

Prog

(PARI) Vec(1818362*x*(1-x)/(1-128405450990*x+x^2) + O(x^20)) \\ Colin Barker, Oct 21 2014
(Magma) I:=[1818362, 233487592671260018]; [n le 2 select I[n] else 128405450990*Self(n-1) - Self(n-2): n in [1..10]]; # G. C. Greubel, Jan 10 2020
(Sage)
def A147525_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 1818362*x*(1-x)/(1-128405450990*x+x^2) ).list()
a=A147525_list(10); a[1:] # G. C. Greubel, Jan 10 2020
(GAP) a:=[1818362, 233487592671260018];; for n in [3..10] do a[n]:=128405450990*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 10 2020

Crossrefs

Cf. A147526

Sequence in context: A137819 A145537 A183735 * A205412 A249211 A244074

Adjacent sequences: A147522 A147523 A147524 * A147526 A147527 A147528

Keyword

easy,nonn

Author

Richard Choulet, Nov 06 2008

Extensions

Editing and a(6) from Colin Barker, Oct 21 2014

Status

approved