OFFSET
0,3
COMMENTS
See A100828 for a similar case.
If the pair (1,1)=(x,y), iteration of x'=3*x+4*y and y'=2*x+3*y gives a new pair of integer satisfying Pell's equation x^2-2*y^2=-1. Example: 7^2-2*5^2=-1; 41^2-2*29^2=-1. [Vincenzo Librandi, Nov 13 2010]
Floretion Algebra Multiplication Program, FAMP Code: 1jesleftcycseq:['k + i' + j']
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..1000
T. W. Forget and T. A. Larkin, Pythagorean triads of the form X, X+1, Z described by recurrence sequences, Fib. Quart., 6 (No. 3, 1968), 94-104.
Morris Newman, Daniel Shanks, and H. C. Williams, Simple groups of square order and an interesting sequence of primes, Acta Arith., 38 (1980/1981) 129-140. MR82b:20022.
The Prime Glossary, NSW number.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).
FORMULA
G.f.: (1+x-x^2+x^3)/((x^2+2*x-1)*(x^2-2*x-1)).
a(n) = ((1+2*sqrt(2)+(-1)^n)*(1+sqrt(2))^n-(1-2*sqrt(2)+(-1)^n)*(1-sqrt(2))^n)/(4*sqrt(2)). [Bruno Berselli, Apr 04 2012]
MATHEMATICA
LinearRecurrence[{0, 6, 0, -1}, {1, 1, 5, 7}, 30] (* Bruno Berselli, Apr 04 2012 *)
PROG
(Maxima) makelist(expand(((1+2*sqrt(2)+(-1)^n)*(1+sqrt(2))^n-(1-2*sqrt(2)+(-1)^n)*(1-sqrt(2))^n)/(4*sqrt(2))), n, 0, 29); /* Bruno Berselli, Apr 04 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Creighton Dement, Apr 22 2005
STATUS
approved