A138288 a(n) = A054320(n) - A001078(n).

1, 9, 89, 881, 8721, 86329, 854569, 8459361, 83739041, 828931049, 8205571449, 81226783441, 804062262961, 7959395846169, 78789896198729, 779939566141121, 7720605765212481, 76426118085983689, 756540575094624409, 7488979632860260401, 74133255753507979601, 733843577902219535609

Offset: 0

Author

Reinhard Zumkeller, Mar 12 2008

Keywords

• nonn
• easy

Comments

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

Examples

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

References

• H. Brocard, Note #2049, L'Intermédiaire des Mathématiciens, 8 (1901), pp. 212-213. - N. J. A. Sloane, Mar 02 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2024.
• 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.
• Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (I).
• Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (II).
• Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (III).
• Editors, L'Intermédiaire des Mathématiciens, Query 4500: The equation x(x+1)/2 = y*(y+1)/3, L'Intermédiaire des Mathématiciens, 22 (1915), 255-260 (IV).
• Index entries for linear recurrences with constant coefficients, signature (10, -1).

Crossrefs

Cf. A001078, A001079, A072256, A138281.

Cf. similar sequences listed in A238379.

Cf. A041007, A041039, A054320.

Programs

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 *)

PARI

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

Sage

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

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.

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

From Michael Somos, Jan 25 2013: (Start)

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

a(-1-n) = a(n). (End)

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

E.g.f.: exp(5*x)*(3*cosh(2*sqrt(6)*x) + sqrt(6)*sinh(2*sqrt(6)*x))/3. - Stefano Spezia, May 16 2023