A104683 Interlaces "2*n^2 - 1 is a square" with NSW numbers.

1, 1, 5, 7, 29, 41, 169, 239, 985, 1393, 5741, 8119, 33461, 47321, 195025, 275807, 1136689, 1607521, 6625109, 9369319, 38613965, 54608393, 225058681, 318281039, 1311738121, 1855077841, 7645370045, 10812186007, 44560482149, 63018038201

Offset: 0

Author

Creighton Dement, Apr 22 2005

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001653, A002315, A100828.

Programs

Mathematica

LinearRecurrence[{0, 6, 0, -1}, {1, 1, 5, 7}, 30] (* Bruno Berselli, Apr 04 2012 *)

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 */

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]