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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: 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

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Last modified September 23 23:21 EDT 2018. Contains 315306 sequences. (Running on oeis4.)