A106329 Numbers k such that k^2 = 8*j^2 + 9.

3, 9, 51, 297, 1731, 10089, 58803, 342729, 1997571, 11642697, 67858611, 395508969, 2305195203, 13435662249, 78308778291, 456417007497, 2660193266691, 15504742592649, 90368262289203, 526704831142569, 3069860724566211, 17892459516254697, 104284896372961971

Offset: 1

Author

Pierre CAMI, Apr 29 2005

Keywords

• nonn
• easy

Comments

The ratio a(n)/(2*j(n)) tends to sqrt(2) as n increases.

After 3, first differences of A301383. - Bruno Berselli, Mar 22 2018

For n > 0, a(n+1) is the n-th almost Lucas-balancing number of first type (see Tekcan and Erdem). - Stefano Spezia, Nov 25 2022

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Tanya Khovanova, Recursive Sequences
• Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.
• Ahmet Tekcan and Alper Erdem, General Terms of All Almost Balancing Numbers of First and Second Type, arXiv:2211.08907 [math.NT], 2022.
• Index entries for linear recurrences with constant coefficients, signature (6,-1).

Crossrefs

Cf. A003499, A103328, A301383.

Programs

Mathematica

CoefficientList[Series[3 x (1 - 3 x)/(1 - 6 x + x^2), {x, 0, 23}], x] (* Michael De Vlieger, Nov 02 2020 *)

PARI

Vec((3-9*x)/(1-6*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Dec 28 2011

Formula

a(1)=3, a(2)=9 then a(n) = 6*a(n-1)-a(n-2).

G.f.: 3*x*(1 - 3*x)/(1 - 6*x + x^2). - Philippe Deléham, Nov 17 2008

a(n) = (3/2)*A003499(n-1).

a(n) = 3*((3-2*sqrt(2))^(n-1) + (3+2*sqrt(2))^(n-1))/2. - Colin Barker, Oct 13 2015

E.g.f.: 3*exp(3*x)*(3*cosh(2*sqrt(2)*x) - 2*sqrt(2)*sinh(2*sqrt(2)*x)) - 9. - Stefano Spezia, Nov 25 2022