A114048 -id:A114048 - cp's OEIS frontend

A167774 Subsequence of A167708 whose indices are congruent to 1 mod 5, i.e., a(n) = A167708(5*n+1).

9, 1530, 520191, 176863410, 60133039209, 20445056467650, 6951259065961791, 2363407637370541290, 803551645446918076809, 273205196044314775573770, 92888963103421576777004991, 31581974249967291789406123170, 10737778356025775786821304872809

Offset: 0

Richard Choulet, Nov 11 2009

			a(0)=A167708(1)=9, a(1)=A167708(6)=1530.

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (340,-1).

Cf. A114048, A167708, A167709, A167775, A167778, A167779, A167780.

I:=[9, 1530]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 24 2016

u(0):=9:for n from 0 to 20 do u(n+1):=170*u(n)+39*sqrt(19*u(n)^2-1539):od:seq(u(n),n=0..20); taylor(((9+1530*z-9*z*340)/(1-340*z+z^2)),z=0,20);

LinearRecurrence[{340, -1}, {9, 1530}, 50] (* G. C. Greubel, Jun 23 2016 *)

Recurrence formulas: a(n+2) = 340*a(n+1) - a(n) or a(n+1) = 170*a(n) + 39*sqrt(19*a(n)^2 - 1539).
G.f.: (9 - 1530*z)/(1 - 340*z + z^2).
a(n) = (9/2)*(170 + 39*sqrt(19))^(n) + (9/2)*(170 - 39*sqrt(19))^(n).
a(n) = 9*A114048(n+1). - R. J. Mathar, Feb 19 2016

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

Original entry on oeis.org

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

Offset: 0

Gary Detlefs, Feb 20 2012

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
   -  -----------------------  -----------------------
A001542 {0,2,6,-1}       A001541 {1,3,6,-1}
A001353 {0,1,4,-1}       A001075 {1,2,4,-1}
A060645 {0,4,18,-1}      A023039 {1,9,18,-1}
A001078 {0,2,10,-1}      A001079 {1,5,10,-1}
A001080 {0,3,16,-1}      A001081 {1,8,16,-1}
A001109 {0,1,6,-1}       A001541 {1,3,6,-1}
A084070 {0,1,38,-1}      A078986 {1,19,38,-1}
A001084 {0,3,20,-1}      A001085 {1,10,20,-1}
A011944 {0,2,14,-1}      A011943 {1,7,14,-1}
A075871 {0,180,1298,-1}  A114047 {1,649,1298,-1}
A068204 {0,4,30,-1}      A069203 {1,15,30,-1}
A001090 {0,1,8,-1}       A001091 {1,4,8,-1}
A121740 {0,8,66,-1}      A099370 {1,33,66,-1}
A202299 {0,4,34,-1}      A056771 {1,17,34,-1}
A174765 {0,39,340,-1}    A114048 {1,179,340,-1}
a(n)    {0,2,18,-1}      A023039 {1,9,18,-1}
A174745 {0,12,110,-1}    A114049 {1,55,110,-1}
A174766 {0,42,394,-1}    A114050 {1,197,394,-1}
A174767 {0,5,48,-1}      A114051 {1,24,48,-1}
A004189 {0,1,10,-1}      A001079 {1,5,10,-1}
A174768 {0,10,102,-1}    A099397 {1,51,102,-1}
The sequence of the c parameter is listed in A180495.

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).

Cf. A023039, A049660, A079962.

m:=16; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // Bruno Berselli, Jun 19 2019

readlib(issqr):for x from 1 to 720016798592704030 do if issqr(20*x^2+1) then print(x) fi od;

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 *)

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

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

A174765 y-values in the solution to x^2 - 19*y^2 = 1.

Original entry on oeis.org

0, 39, 13260, 4508361, 1532829480, 521157514839, 177192022215780, 60244766395850361, 20483043382566906960, 6964174505306352516039, 2367798848760777288546300, 805044644404158971753225961

Offset: 1

Vincenzo Librandi, Apr 14 2010

The corresponding values of x of this Pell equation are in A114048.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (340,-1).

Cf. A114048.

I:=[0, 39]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..20]];

Mathematica
```
LinearRecurrence[{340,-1},{0,39},30]
```

a(n) = 340*a(n-1)-a(n-2) with a(1)=0, a(2)=39.

G.f.: 39*x^2/(1-340*x+x^2).

cp's OEIS Frontend

A167774 Subsequence of A167708 whose indices are congruent to 1 mod 5, i.e., a(n) = A167708(5*n+1).

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

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A174765 y-values in the solution to x^2 - 19*y^2 = 1.

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