OFFSET
1,1
COMMENTS
Lim_{n->infinity} a(n)/a(n-1) = 3 + 2*sqrt(2).
Also gives solutions to the equation x^2-2 = floor(x*r*floor(x/r)) where r=sqrt(2). - Benoit Cloitre, Feb 14 2004
The upper intermediate convergents to 2^(1/2) beginning with 10/7, 58/41, 338/239, 1970/1393 form a strictly decreasing sequence; essentially, numerators = A075870, denominators = A002315. - Clark Kimberling, Aug 27 2008
Numbers n such that sqrt(floor(n^2/2 - 1)) is an integer. The integer square roots are given by A002315. - Richard R. Forberg, Aug 01 2013
a(n) are the integer square roots of m^2 + (m+2)^2. The values of m are given by A065113 (except for m = 0). The values of this expression are given by A165518. - Richard R. Forberg, Aug 15 2013
Values of x (or y) in the solutions to x^2 - 6*x*y + y^2 + 16 = 0. - Colin Barker, Feb 04 2014
Also integers k such that k^2 is equal to the sum of four consecutive triangular numbers. - Colin Barker, Dec 20 2014
Equivalently, numbers x such that (x-1)*x/2 + x*(x+1)/2 = (y-1)^2 + (y+1)^2. y-values are listed in A002315. Example: for x=58 and y=41, 57*58/2 + 58*59/2 = 40^2 + 42^2. - Bruno Berselli, Mar 19 2018
REFERENCES
A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.
P.-F. Teilhet, Reply to Query 2094, L'Intermédiaire des Mathématiciens, 10 (1903), 235-238. - N. J. A. Sloane, Mar 03 2022
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
Index entries for linear recurrences with constant coefficients, signature (6,-1).
FORMULA
a(n) = 2 * A001653(n).
a(n) = (1/sqrt(2))*((1+sqrt(2))^(2*n-1) - (1-sqrt(2))^(2*n-1)) = 6*a(n-1) - a(n-2).
G.f.: 2*x*(1-x)/(1-6*x+x^2). - Philippe Deléham, Nov 17 2008
a(n) = round(((2+sqrt(2))*(3+2*sqrt(2))^(n-1))/2). - Paul Weisenhorn, Jun 11 2020
From Peter Bala, Aug 21 2022: (Start)
a(n) = 2*Pell(2*n-1).
1/a(n) - 1/a(n+1) = 1/(Pell(2*n) + 1/Pell(2*n)), where Pell(n) = A000129(n). (End)
EXAMPLE
From Muniru A Asiru, Mar 19 2018: (Start)
For k=2, 2*2^2 - 4 = 8 - 4 = 4 = 2^2.
For k=10, 2*10^2 - 4 = 200 - 4 = 196 = 14^2.
For k=58, 2*58^2 - 4 = 6728 - 4 = 6724 = 82^2.
... (End)
MAPLE
a:= proc(n) option remember: if n = 1 then 2 elif n = 2 then 10 elif n >= 3 then 6*procname(n-1) - procname(n-2) fi; end: seq(a(n), n = 0..25); # Muniru A Asiru, Mar 19 2018
MATHEMATICA
LinearRecurrence[{6, -1}, {2, 10}, 30] (* Harvey P. Dale, Sep 27 2018 *)
PROG
(PARI) Vec(2*x*(1-x)/(1-6*x+x^2) + O(x^100)) \\ Colin Barker, Dec 20 2014
(GAP) a:=[2, 10];; for n in [3..25] do a[n]:=6*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Mar 19 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gregory V. Richardson, Oct 16 2002
EXTENSIONS
More terms from Colin Barker, Dec 20 2014
STATUS
approved