Search: id:A077232
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%I A077232
%S A077232 1,2,2,5,8,3,3,10,7,18,15,4,4,17,170,9,55,197,24,5,5,26,127,70,11,1520,
%T A077232 17,23,35,6,6,37,25,19,32,13,3482,199,161,24335,48,7,7,50,649,182,485,
%U A077232 89,15,151,99,530,31,29718,63,8,8,65,48842,33,7775,251,3480,17,1068,43,
               26,57799,351,53,80,9,9,82,55,378,10405,28,197,500,19,1574,1151,12151,
               2143295,39,49,5604,99,10,10,101,227528
%N A077232 a(n) is smallest natural number satisfying Pell equation a^2 - d(n)*b^2= 
               +1 or = -1, with d(n)=A000037(n) (a nonsquare). Corresponding smallest 
               b(n)=A077233(n).
%C A077232 If d(n)=A000037(n) is from A003654 (that is if the regular continued 
               fraction for sqrt(d(n)) has odd (primitive) period length) then the 
               -1 option applies. For such d(n) the minimal a(n) and b(n) numbers 
               for the +1 option are 2*a(n)^2+1 and 2*a(n)*b(n), respectively (see 
               Perron I, pp. 94,95).
%C A077232 If d(n)=A000037(n)= k^2+1, k=1,2,.., then the a^2 - d(n)*b^2 = -1 Pell 
               equation has the minimal solution a(n)=k and b(n)=1. If d(n)=A000037(n)= 
               k^2-1, k=2,3,..., then the a^2 - d(n)*b^2 = +1 Pell equation has 
               the minimal solution a=k and b=1.
%C A077232 The general integer solutions (up to signs) of Pell equation a^2 - d(n)*b^2 
               = +1 with d(n)=A000037(n), but not from A003654, are a(n,p)= T(p+1,
               a(n)) and b(n,p)= b(n)*S(p,2*a(n)), p=0,1,... If d(n)=A000037(n) 
               is also from A003654 then these solutions are a(n,p)= T(p+1,2*a(n)^2+1) 
               and b(n,p)= 2*a(n)*b(n)*S(p,2*(2*a(n)^2+1)), p=0,1,... Here T(n,x), 
               resp. S(n,x) := U(n,x/2), are Chebyshev's polynomials of the first, 
               resp. second, kind. See A053120 and A049310.
%C A077232 The general integer solutions (up to signs) of the Pell equation a^2 
               - d(n)*b^2 = -1 with d(n)=A000037(n)= A003654(k), for some k>=1, 
               are a(n,p) = a(n)*(S(n,2*(2*a(n)^2)+1) + S(n-1,2*(2*a(n)^2)+1)) and 
               b(n,p) = b(n)*(S(n,2*(2*a(n)^2)+1) - S(n-1,2*(2*a(n)^2)+1)) with 
               the S(n,x) := U(n,x/2) Chebyshev polynomials. S(-1,x) := 0.
%C A077232 If the trivial solution x=1, y=0 is included, the sequence becomes A006702. 
               - T. D. Noe (noe(AT)sspectra.com), May 17 2007
%D A077232 T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, 
               table p. 301.
%D A077232 O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 
               (Sec. 26, p. 91 with explanation on pp. 94,95).
%H A077232 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A077232 a(n)=sqrt(A000037(n)*A077233(n)^2 + (-1)^(c(n))) with c(n)=1 if A000037(n)=A003654(k) 
               for some k>=1 else c(n)=0.
%e A077232 d=10=A000037(7)=A003654(3), therefore a(7)^2=10*b(7)^2 -1, i.e. 3^2=10*1^2 
               -1 and 2*a(7)^2+1=19 and 2*a(7)*b(7)=2*3*1=6 satisfy 19^2 - 10*6^2 
               = +1.
%e A077232 d=11=A000037(8) is not in A003654, therefore there is no (nontrivial) 
               solution of the a^2 - d*b^2 = -1 Pell equation and a(8)=10 and b(8)=A077233(8)=3 
               satisfy 10^2 - 11*3^2 = +1.
%e A077232 10=d(7)=A000037(7)=A003654(3)=3^2+1 hence a(7)=3 and b(7)=1 are the smallest 
               numbers satisfying a^2-10*b^2=-1.
%e A077232 8=d(6)=A000037(6)=3^2-1 (not in A003654) hence a(6)=3 and b(6)=1 are 
               the smallest numbers satisfying a^2-8*b^2=+1.
%Y A077232 Cf. A033313, A003814.
%Y A077232 Sequence in context: A038750 A074476 A011021 this_sequence A087910 A035570 
               A126291
%Y A077232 Adjacent sequences: A077229 A077230 A077231 this_sequence A077233 A077234 
               A077235
%K A077232 nonn,nice
%O A077232 1,2
%A A077232 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 
               2002

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