A157623 781250n^2 - 455000n + 66249.

392499, 2281249, 5732499, 10746249, 17322499, 25461249, 35162499, 46426249, 59252499, 73641249, 89592499, 107106249, 126182499, 146821249, 169022499, 192786249, 218112499, 245001249, 273452499, 303466249, 335042499, 368181249

Offset

1,1

Comments

The identity (781250*n^2-455000*n+66249)^2-(625*n^2-364*n+53)*(31250*n-9100)^2=1 can be written as a(n)^2-A157621(n)*A157622(n)^2=1.

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(1)=392499, a(2)=2281249, a(3)=5732499, a(n)=3*a(n-1)-3*a(n-2)+ a(n-3) [From Harvey P. Dale, Jul 11 2011]

G.f.: (392499+1103752*x+66249*x^2)/(1-x)^3 [From Harvey P. Dale, Jul 11 2011]

Example

For n=1, a(1)=392499; n=2, a(2)=2281249; n=3, a(3)=5732499.

Mathematica

Table[781250n^2-455000n+66249, {n, 25}] (* or *) LinearRecurrence[{3, -3, 1}, {392499, 2281249, 5732499}, 25] (* or *) CoefficientList[Series[ (-392499- 1103752 x-66249 x^2)/(x-1)^3, {x, 0, 25}], x] (* Harvey P. Dale, Jul 11 2011 *)

Prog

(Magma) I:=[392499, 2281249, 5732499]; [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)=781250*n^2-455000*n+66249

Crossrefs

Cf. A157621, A157622.

Sequence in context: A346340 A345638 A346351 * A145228 A204628 A171439

Adjacent sequences: A157620 A157621 A157622 * A157624 A157625 A157626

Keyword

nonn,easy

Author

Vincenzo Librandi, Mar 03 2009

Extensions

Minor corrections by M. F. Hasler, Oct 08 2014

Status

approved