A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.

0, 3, 18, 105, 612, 3567, 20790, 121173, 706248, 4116315, 23991642, 139833537, 815009580, 4750223943, 27686334078, 161367780525, 940520349072, 5481754313907, 31950005534370, 186218278892313, 1085359667819508, 6325939728024735, 36870278700328902, 214895732473948677

Offset

1,2

Comments

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

The squares of the numbers in this sequence are one less than a triangular number: a(n)^2 = A164080(n). For example, 18^2 is 324, and 325 is a triangular number. a(n)^2 + 1 = A164055(n). a(n)^2 = A072221(n)(A072221(n)+1)/2 - 1. - Tanya Khovanova & Alexey Radul, Aug 09 2009

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

Links

Reinhard Zumkeller, 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).

Formula

a(1)=0, a(2)=3 then a(n) = 6*a(n-1) - a(n-2).

a(n) = ((3+2*sqrt(2))^(n-1) - (3-2*sqrt(2))^(n-1))*3/4/sqrt(2). - Max Alekseyev, Jan 11 2007

a(n) = 3*A001109(n). - M. F. Hasler, R. J. Mathar, Jun 03 2009

a(n) = (3/4)*A005319(n-1).

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

E.g.f.: 3 - 3*exp(3*x)*(4*cosh(2*sqrt(2)*x) - 3*sqrt(2)*sinh(2*sqrt(2)*x))/4. - Stefano Spezia, Nov 25 2022

Mathematica

s=0; lst={}; Do[s+=n; If[Sqrt[s-1]==Floor[Sqrt[s-1]], AppendTo[lst, Sqrt[s-1]]], {n, 8!}]; lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Rest@ CoefficientList[Series[3 x^2/(1 - 6 x + x^2), {x, 0, 24}], x] (* Michael De Vlieger, Nov 02 2020 *)

Prog

(Haskell)
a106328 n = a106328_list !! (n-1)
a106328_list = 0 : 3 : zipWith (-) (map (* 6) (tail a106328_list)) a106328_list
-- Reinhard Zumkeller, Jan 10 2012
(PARI) concat(0, Vec(3*x^2/(1-6*x+x^2) + O(x^40))) \\ Michel Marcus, Sep 07 2016
(PARI) a(n)=([0, 1; -1, 6]^n*[-3; 0])[1, 1] \\ Charles R Greathouse IV, Sep 07 2016

Crossrefs

Cf. A001109, A005319, A072221, A103328, A164080, A164055.

Sequence in context: A303519 A124408 A136779 * A007277 A025595 A151331

Adjacent sequences: A106325 A106326 A106327 * A106329 A106330 A106331

Keyword

nonn,easy

Author

Pierre CAMI, Apr 29 2005

Extensions

More terms from Max Alekseyev, Jan 11 2007

Status

approved