A207832 Numbers x such that 20*x^2 + 1 is a perfect square.

0, 2, 36, 646, 11592, 208010, 3732588, 66978574, 1201881744, 21566892818, 387002188980, 6944472508822, 124613502969816, 2236098580947866, 40125160954091772, 720016798592704030

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<x>:=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