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

Author

Gary Detlefs, Feb 20 2012

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A023039, A049660, A079962.

Programs

Magma

m:=16; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // 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 *)
Table[2 ChebyshevU[-1 + n, 9], {n, 0, 16}]  (* Herbert Kociemba, Jun 05 2022 *)

Maxima

makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15); /* Bruno Berselli, Jun 19 2019 */

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