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A104683 Interlaces "2*n^2 - 1 is a square" with NSW numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
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]

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, 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

Floretion Algebra Multiplication Program, FAMP Code: 1jesleftcycseq:['k + i' + j']

(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

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Last modified February 25 12:01 EST 2020. Contains 332233 sequences. (Running on oeis4.)