A084232 RMS values associated with A084231.

1, 195, 37829, 7338631, 1423656585, 276182038859, 53577891882061, 10393834843080975, 2016350381665827089, 391161580208327374291, 75883330210033844785365, 14720974899166357560986519, 2855793247108063332986599321, 554009168964065120241839281755

Offset: 0

Author

Ignacio Larrosa Cañestro, May 20 2003

Keywords

• nonn
• easy
• changed

Comments

From Klaus Purath, Aug 20 2025: (Start)

Solutions to the Pell equation (7*b(n))^2 - 3*(4*a(n))^2 = 1. The corresponding b(n) are given by A302332.

For any two consecutive terms (x,y), x^2 - 194*x*y + y^2 - 196 = 0. By analogy to this, for three consecutive terms (x, y, z), y^2 - x*z - 196 = 0. (End)

Examples

			a(1)=195 because 195 = sqrt((Sum_{k=1..337}k^2)/337) and 337 = A084231(1).
		

Links

• Indranil Ghosh, Table of n, a(n) for n = 0..436
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (194,-1).

Crossrefs

Cf. A084231, A217855, A302332.

Programs

Mathematica

LinearRecurrence[{194,-1},{1,195},20] (* Harvey P. Dale, Nov 10 2021 *)

Formula

a(n) = ((7+4*sqrt(3))^(2*n+1)-(7-4*sqrt(3))^(2*n+1))/(8*sqrt(3)). [simplified by Bruno Berselli, Oct 19 2012]

a(n) = floor(((7*sqrt(3) + 12)/24)*(56*sqrt(3) + 97)^n).

a(n+2) = 194*a(n+1) - a(n).

G.f.: (1-x)/(1-194*x+x^2). - Philippe Deléham, Nov 18 2008

a(n)^2 = (Sum_{i=1..A084231(n+1)}i^2)/A084231(n+1). - Bruno Berselli, Oct 17 2012