|
| |
|
|
A075870
|
|
2*n^2 - 4 is a square.
|
|
4
| |
|
|
2, 10, 58, 338, 1970, 11482, 66922, 390050, 2273378, 13250218, 77227930, 450117362, 2623476242, 15290740090, 89120964298, 519435045698, 3027489309890, 17645500813642, 102845515571962, 599427592618130
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,1
|
|
|
COMMENTS
| Lim. n-> Inf. 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 (benoit7848c(AT)orange.fr), 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 (ck6(AT)evansville.edu), Aug 27 2008
|
|
|
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, p. 341-400.
Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, p. 139-147.
|
|
|
LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
J. J. O'Connor and E. F. Robertson, Pell's Equation
Eric Weisstein's World of Mathematics, Pell Equation.
|
|
|
FORMULA
| 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.: 2x(1-x)/(1-6x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 17 2008]
|
|
|
CROSSREFS
| Cf. 2*A001653.
Sequence in context: A191277 A093303 A199163 * A074608 A086871 A108450
Adjacent sequences: A075867 A075868 A075869 * A075871 A075872 A075873
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Gregory V. Richardson (omomom(AT)hotmail.com), Oct 16 2002
|
| |
|
|
