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A207832 20*x^2+1 is a perfect square. 1
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).

G.f.: 2*x/(1-18*x+x^2). - Bruno Berselli, Feb 21 2012

a(n) = -a(-n) = 2*A049660(n) = ((2+sqrt(5))^(2n)-(2-sqrt(5))^(2n))/(4*sqrt(5)). - Bruno Berselli, Feb 21 2012

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<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2)));

(Maxima) makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15);

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 June 22 23:15 EDT 2017. Contains 288633 sequences.