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

0,3

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

Cf. A001653, A002315, A100828.

Sequence in context: A213901 A087901 A018776 * A153121 A280926 A070153

Adjacent sequences: A104680 A104681 A104682 * A104684 A104685 A104686

Keyword

nonn,easy

Author

Creighton Dement, Apr 22 2005

Status

approved