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%I #35 Mar 06 2024 16:40:05
%S 1,1,5,7,29,41,169,239,985,1393,5741,8119,33461,47321,195025,275807,
%T 1136689,1607521,6625109,9369319,38613965,54608393,225058681,
%U 318281039,1311738121,1855077841,7645370045,10812186007,44560482149,63018038201
%N Interlaces "2*n^2 - 1 is a square" with NSW numbers.
%C See A100828 for a similar case.
%C 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]
%C Floretion Algebra Multiplication Program, FAMP Code: 1jesleftcycseq:['k + i' + j']
%D A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
%H Bruno Berselli, <a href="/A104683/b104683.txt">Table of n, a(n) for n = 0..1000</a>
%H T. W. Forget and T. A. Larkin, <a href="http://www.fq.math.ca/Scanned/6-3/forget.pdf">Pythagorean triads of the form X, X+1, Z described by recurrence sequences</a>, Fib. Quart., 6 (No. 3, 1968), 94-104.
%H Morris Newman, Daniel Shanks, and H. C. Williams, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa38/aa3826.pdf">Simple groups of square order and an interesting sequence of primes</a>, Acta Arith., 38 (1980/1981) 129-140. MR82b:20022.
%H The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=NSWNumber">NSW number.</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F G.f.: (1+x-x^2+x^3)/((x^2+2*x-1)*(x^2-2*x-1)).
%F 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]
%t LinearRecurrence[{0, 6, 0, -1}, {1, 1, 5, 7}, 30] (* _Bruno Berselli_, Apr 04 2012 *)
%o (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 */
%Y Cf. A001653, A002315, A100828.
%K nonn,easy
%O 0,3
%A _Creighton Dement_, Apr 22 2005
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