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A138288 a(n) = A054320(n) - A001078(n). 10
1, 9, 89, 881, 8721, 86329, 854569, 8459361, 83739041, 828931049, 8205571449, 81226783441, 804062262961, 7959395846169, 78789896198729, 779939566141121, 7720605765212481, 76426118085983689, 756540575094624409 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

(sqrt(2)+sqrt(3))^(2*n+1) = A054320(n-1)*sqrt(2) + a(n)*sqrt(3).

Numbers n such that 6*n^2-2 is a square. [Bruno Berselli, Feb 10 2014]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Bruno Deschamps, Sur les bonnes valeurs initiales de la suite de Lucas-Lehmer, Journal of Number Theory, Volume 130, Issue 12, December 2010, Pages 2658-2670.

FORMULA

a(n) = A072256(n+1).

a(n) = A001079(n) + 2*A001078(n).

a(n) = 10*a(n-1) - a(n-2). a(-1) = a(0) = 1.

G.f.: (1 - x) / (1 - 10*x + x^2). a(-1-n) = a(n). - Michael Somos, Jan 25 2013

a(n) = sqrt(2+(5-2*sqrt(6))^(1+2*n)+(5+2*sqrt(6))^(1+2*n))/(2*sqrt(3)). - Gerry Martens, Jun 04 2015

EXAMPLE

1 + 9*x + 89*x^2 + 881*x^3 + 8721*x^4 + 86329*x^5 + ...

MATHEMATICA

CoefficientList[Series[(1 - x)/(1 - 10 x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)

a[c_, n_] := Module[{},

  p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];

  d := Denominator[Convergents[Sqrt[c], n p]];

  t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];

  Return[t];

  ] (* Complement of A041007, A041039 *)

a[6, 20] (* Gerry Martens, Jun 07 2015 *)

PROG

(Sage) [lucas_number1(n, 10, 1)-lucas_number1(n-1, 10, 1) for n in xrange(1, 20)] # Zerinvary Lajos, Nov 10 2009

(PARI) {a(n) = subst( poltchebi(n+1) + poltchebi(n), x, 5) / 6} /* Michael Somos, Jan 25 2013 */

CROSSREFS

Cf. A001078, A001079, A072256, A138281.

Cf. similar sequences listed in A238379.

Cf. A041007, A041039.

Sequence in context: A320093 A015584 A072256 * A059482 A109002 A142991

Adjacent sequences:  A138285 A138286 A138287 * A138289 A138290 A138291

KEYWORD

nonn,easy

AUTHOR

Reinhard Zumkeller, Mar 12 2008

STATUS

approved

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Last modified December 13 22:07 EST 2018. Contains 318087 sequences. (Running on oeis4.)