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A106328 Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k. 11
0, 3, 18, 105, 612, 3567, 20790, 121173, 706248, 4116315, 23991642, 139833537, 815009580, 4750223943, 27686334078, 161367780525, 940520349072, 5481754313907, 31950005534370, 186218278892313, 1085359667819508, 6325939728024735, 36870278700328902, 214895732473948677 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Tanya Khovanova, Recursive Sequences

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

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 *)

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. A103328, A164080, A164055, A072221.

Sequence in context: A009021 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

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Last modified June 23 02:40 EDT 2017. Contains 288633 sequences.