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 A207832 Numbers x such that 20*x^2 + 1 is a perfect square. 4
 0, 2, 36, 646, 11592, 208010, 3732588, 66978574, 1201881744, 21566892818, 387002188980, 6944472508822, 124613502969816, 2236098580947866, 40125160954091772, 720016798592704030 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Denote as {a,b,c,d} the second-order linear recurrence a(n) = c*a(n-1) + d*a(n-2) with initial terms a, b. The following sequences and recurrence formulas are related to integer solutions of k*x^2 + 1 = y^2. . k x y - ----------------------- ----------------------- 2 A001542 {0,2,6,-1} A001541 {1,3,6,-1} 3 A001353 {0,1,4,-1] A001075 {1,2,4,-1} 5 A060645 {0,4,18,-1} A023039 {1,9,18,-1} 6 A001078 {0,2,10,-1} A001079 {1,5,10,-1} 7 A001080 {0,3,16,-1} A001081 {1,8,16,-1} 8 A001109 {0,1,6,-1} A001541 {1,3,6,-1} 10 A084070 {0,1,38,-1} A078986 {1,19,38,-1} 11 A001084 {0,3,20,-1} A001085 {1,10,20,-1} 12 A011944 {0,2,14,-1} A011943 {1,7,14,-1} 13 A075871 {0,180,1298,-1} A114047 {1,649,1298,-1} 14 A068204 {0,4,30,-1} A069203 {1,15,30,-1} 15 A001090 {0,1,8,-1} A001091 {1,4,8,-1} 17 A121740 {0,8,66,-1} A099370 {1,33,66,-1} 18 A202299 {0,4,34,-1} A056771 {1,17,34,-1} 19 A174765 {0,39,340,-1} A114048 {1,179,340,-1} 20 a(n) {0,2,18,-1} A023039 {1,9,18,-1} 21 A174745 {0,12,110,-1} A114049 {1,55,110,-1} 22 A174766 {0,42,394,-1} A114050 {1,197,394,-1} 23 A174767 {0,5,48,-1} A114051 {1,24,48,-1} 24 A004189 {0,1,10,-1} A001079 {1,5,10,-1} 26 A174768 {0,10,102,-1} A099397 {1,51,102,-1} The sequence of the c parameter is listed in A180495. LINKS Bruno Berselli, Table of n, a(n) for n = 0..500 Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5. Index entries for linear recurrences with constant coefficients, signature (18,-1). FORMULA a(n) = 18*a(n-1) - a(n-2). From Bruno Berselli, Feb 21 2012: (Start) G.f.: 2*x/(1-18*x+x^2). a(n) = -a(-n) = 2*A049660(n) = ((2 + sqrt(5))^(2*n)-(2 - sqrt(5))^(2*n))/(4*sqrt(5)). (End) a(n) = Fibonacci(6*n)/4. - Bruno Berselli, Jun 19 2019 For n>=1, a(n) = A079962(6n-3). - Christopher Hohl, Aug 22 2021 MAPLE readlib(issqr):for x from 1 to 720016798592704030 do if issqr(20*x^2+1) then print(x) fi od; MATHEMATICA LinearRecurrence[{18, -1}, {0, 2}, 16] (* Bruno Berselli, Feb 21 2012 *) Table[2 ChebyshevU[-1 + n, 9], {n, 0, 16}] (* Herbert Kociemba, Jun 05 2022 *) PROG (Magma) m:=16; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // Bruno Berselli, Jun 19 2019 (Maxima) makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15); /* Bruno Berselli, Jun 19 2019 */ CROSSREFS Cf. A023039, A049660, A079962. Sequence in context: A228790 A124104 A262973 * A112036 A336714 A093530 Adjacent sequences: A207829 A207830 A207831 * A207833 A207834 A207835 KEYWORD nonn,easy AUTHOR Gary Detlefs, Feb 20 2012 STATUS approved

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Last modified November 30 21:14 EST 2023. Contains 367462 sequences. (Running on oeis4.)