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) = (3/4)*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
KEYWORD
nonn,easy
AUTHOR
Pierre CAMI, Apr 29 2005
EXTENSIONS
More terms from Max Alekseyev, Jan 11 2007
STATUS
approved