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 A207832 Numbers x such that 20*x^2+1 is a perfect square. 2
 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 2  x = A001542 {0,2,6,-1}     y = A001541 {1,3,6,-1} 3  x = A001353 {0,1,4,-1]     y = A001075 {1,2,4,-1} 5  x = A060645 {0,4,18,-1}    y = A023039 {1,9,18,-1} 6  x = A001078 {0,2,10,-1}    y = A001079 {1,5,10,-1} 7  x = A001080 {0,3,16,-1}    y = A001081 {1,8,16,-1} 8  x = A001109 {0,1,6,-1}     y = A001541 {1,3,6,-1} 10 x = A084070 {0,1,38,-1}    y = A078986 {1,19,38,-1} 11 x = A001084 {0,3,20,-1}    y = A001085 {1,10,20,-1} 12 x = A011944 {0,2,14,-1}    y = A011943 {1,7,14,-1} 13 x = A075871 {0,180,1298,-1}y = A114047 {1,649,1298,-1} 14 x = A068204 {0,4,30,-1}    y = A069203 {1,15,30,-1} 15 x = A001090 {0,1,8,-1}     y = A001091 {1,4,8,-1} 17 x = A121740 {0,8,66,-1}    y = A099370 {1,33,66,-1} 18 x = A202299 {0,4,34,-1}    y = A056771 {1,17,34,-1} 19 x = A174765 {0,39,340,-1}  y = A114048 {1,179,340,-1} 20 x = a(n)    {0,2,18,-1}    y = A023039 {1,9,18,-1} 21 x = A174745 {0,12,110,-1}  y = A114049 {1,55,110,-1} 22 x = A174766 {0,42,394,-1}  y = A114050 {1,197,394,-1} 23 x = A174767 {0,5,48,-1}    y = A114051 {1,24,48,-1} 24 x = A004189 {0,1,10,-1}    Y = A001079 {1,5,10,-1} 26 x = A174768 {0,10,102,-1}  y = 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 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 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 *) 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 Sequence in context: A228790 A124104 A262973 * A112036 A093530 A001626 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 October 14 12:02 EDT 2019. Contains 328004 sequences. (Running on oeis4.)