A011944 a(n) = 14*a(n-1) - a(n-2) with a(0) = 0, a(1) = 2.

0, 2, 28, 390, 5432, 75658, 1053780, 14677262, 204427888, 2847313170, 39657956492, 552364077718, 7693439131560, 107155783764122, 1492487533566148, 20787669686161950, 289534888072701152

Offset: 0

Author

E. K. Lloyd

Keywords

• nonn
• easy

Comments

Standard deviation of A011943.

Product x*y, where the pair (x, y) solves for x^2 - 3y^2 = 1, i.e., a(n)=A001075(n)*A001353(n). - Lekraj Beedassy, Jul 13 2006

Solutions m to the Diophantine equation where square m^2 = k*(k+1)/3, corresponding solutions k are in A007654. - Bernard Schott, Apr 10 2021

All solutions for y in Pell equation x^2 - 12*y^2 = 1. Corresponding values for x are in A011943. - Herbert Kociemba, Jun 05 2022

Links

• Tanya Khovanova, Recursive Sequences
• E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243.
• Index entries for linear recurrences with constant coefficients, signature (14,-1).

Crossrefs

a(n) = 2 * A007655 = {A001353(2n)}/2. Cf. A011943.

Cf. A007654.

Programs

Mathematica

LinearRecurrence[{14,-1},{0,2},20] (* Harvey P. Dale, Oct 17 2019 *)
Table[2 ChebyshevU[-1 + n, 7], {n, 0, 18}] (* Herbert Kociemba, Jun 05 2022 *)

Formula

For all members x of the sequence, 12*x^2 +1 is a square. Lim_{n->infinity} a(n)/a(n-1) = 7 + sqrt(12). - Gregory V. Richardson, Oct 13 2002

a(n) = ((7+2*sqrt(12))^(n-1) - (7-2*sqrt(12))^(n-1)) / (2*sqrt(12)). - Gregory V. Richardson, Oct 13 2002

a(n) = 13*(a(n-1) + a(n-2)) - a(n-3). a(n) = 15*(a(n-1) - a(n-2)) + a(n-3). - Mohamed Bouhamida, Sep 20 2006

a(n) = sinh(2n*arcsinh(sqrt(3)))/sqrt(12). - Herbert Kociemba, Apr 24 2008

G.f.: 2x/(1-14*x+x^2). - Philippe Deléham, Nov 17 2008