NSW nos

[N. J. A. Sloane]

David Wilson asked for the reason for the name.
Here is the entry:
I think the Newman-Shanks-Williams reference speaks for itself!
NJAS

%I A002315 M4423 N1869
%S A002315 1,7,41,239,1393,8119,47321,275807,1607521,9369319,54608393,318281039,
%T A002315 1855077841,10812186007,63018038201,367296043199,2140758220993,
%U A002315 12477253282759,72722761475561,423859315570607,2470433131948081
%N A002315 NSW numbers: a(n) = 6*a(n-1) - a(n-2); also a(n)^2 - 2*b(n)^2 = -1 with b(n)=A001653(n).
%D A002315 E. Barcucci et al., A combinatorial interpretation of the recurrence f_{n+1} = 6 f_n - f_{n-1}, Discrete Math., 190 (1998), 235-240.
%D A002315 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256.
%D A002315 A. S. Fraenkel, Recent results and questions in combinatorial game complexities, Theoretical Computer Science, vol. 249, no. 2 (2000), 265-288.
%D A002315 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.
%D A002315 Newman, Morris; Shanks, Daniel; Williams, H. C.; Simple groups of square order and an interesting sequence of primes. Acta Arith. 38 (1980/81), no. 2, 129-140. MR82b:20022
%D A002315 Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
%D A002315 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 288.
%D A002315 P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238.
%H A002315 A. S. Fraenkel, <a href="ftp://ftp.wisdom.weizmann.ac.il/pub/fraenkel/ans1.ps">Arrays, numeration systems and games.</a>
%H A002315 Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=NSWNumber">NSW number.</a>
%H A002315 R. A. Sulanke, <a href="http://www.math.uwaterloo.ca/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A002315 R. A. Sulanke, <a href="http://www.math.uwaterloo.ca/JIS/index.html">Moments of generalized Motzkin paths</a>, J. Integer Sequences, Vol. 3 (2000), #00.1.
%H A002315 E. W. Weisstein, <a href="http://mathworld.wolfram.com/NSWNumber.html">Link to a section of The World of Mathematics.</a>
%H A002315 <a href="http://www.research.att.com/~njas/sequences/Sindx_Ch.html#Cheby"> Index entries for sequences related to Chebyshev polynomials.</a>
%F A002315 a(n) = 1/2((1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1)).
%F A002315 a(n)=sqrt(2*(A001653(n))^2-1). G.f.: (1+x)/(1-6*x+x^2).
%F A002315 a(n)= S(n,6)+S(n-1,6) = S(2*n,sqrt(8)), S(n,x)=U(n,x/2) are Chebyshev's polynomials of the 2nd kind. Cf. A049310. S(n,6)= A001109(n+1).
%F A002315 a(n) ~ 1/2*(sqrt(2) + 1)^(2*n+1) - Joe Keane (jgk at jgk.org), May 15 2002
%F A002315 Lim n -> inf. a(n)/a(n-1) = 3 + 2*Sqrt(2). - Gregory V. Richardson (omomom at hotmail.com), Oct 06 2002
%F A002315 a(n) = [2*[(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] - 5*[(3+2*Sqrt(2))^(n-1) - (3- 2*Sqrt(2))^(n-1)] + [(3+2*Sqrt(2))^(n-2) - (3-2*Sqrt(2))^(n-2)] ] / (4*Sqrt(2)) - Gregory V. Richardson (omomom at hotmail.com), Oct 06 2002
%F A002315 Let q(n,x)=sum(i=0,n,x^(n-i)*binomial(2*n-i,i)); then (-1)^n*q(n,-8)=a(n) - Benoit Cloitre (abcloitre at wanadoo.fr), Nov 10 2002
%o A002315 (PARI) a(n)=if(n<0,0,subst(poltchebi(n+1)-poltchebi(n),x,3)/2)
%o A002315 (PARI) a(n)=if(n<0,0,polsym(x^2-2*x-1,2*n+1)[2*n+2]/2)
%o A002315 (PARI) a(n)=local(w=quadgen(8),u=3+2*w,v=3-2*w);if(n<0,0,simplify(((u^n-v^n)+(u^(n+1)-v^(n+1)))/(4*w)))
%Y A002315 Bisection of A001333. Cf. A001109, A001653.
%K A002315 nonn,easy,nice
%O A002315 0,2
%A A002315 njas
%E A002315 More terms from James A. Sellers (sellersj at math.psu.edu), Feb 16 2000