cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A114047 x such that x^2 - 13*y^2 = 1.

Original entry on oeis.org

1, 649, 842401, 1093435849, 1419278889601, 1842222905266249, 2391203911756701601, 3103780835237293411849, 4028705132934095091878401, 5229256158767620191964752649, 6787570465375238075075157060001, 8810261234800900253827361899128649
Offset: 0

Views

Author

Cino Hilliard, Feb 01 2006

Keywords

Comments

A Pellian equation (Pell's equation). - Benoit Cloitre, Feb 03 2006
Numbers n such that 13*(n^2-1) is a square. - Vincenzo Librandi, Nov 13 2010
The corresponding values y of the solutions of this Pell equation are given in A075871(n). - Wolfdieter Lang, Jun 27 2013

Examples

			(649^2-1)/13 = 180^2.

Links

Crossrefs

Cf. A202155, A075871, A132644.

Programs

  • Magma
    I:=[1,649]; [n le 2 select I[n] else 1298*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jun 14 2015
  • Mathematica
    LinearRecurrence[{1298,-1},{1,649},20] (* or *) With[{c=180Sqrt[13]}, Simplify[Table[1/2((649-c)^n+(649+c)^n),{n,0,20}]]] (* Harvey P. Dale, Aug 11 2011 *)
  • PARI
    /* This sequence is computed with g(1e9,13) in the following program. */
g(n,k) = for(y=0,n,x=k*y^2+1;if(issquare(x),print1(floor(sqrt(x))",")))
  • PARI
    a0=1;a1=649;for(n=2,30,a2=1298*a1-a0;a0=a1;a1=a2;print1(a2,",")) \\ Benoit Cloitre
  • PARI
    Vec((1-649*x)/(1-1298*x+x^2) + O(x^100)) \\ Colin Barker, Jun 13 2015

Formula

a(0)=1, a(1)=649 then a(n)=1298*a(n-1)-a(n-2). - Benoit Cloitre, Feb 03 2006
G.f.: (1-649*x)/(1-1298*x+x^2). - Philippe Deléham, Nov 18 2008
a(n) = 2*A132644(n) + 1. - Hugo Pfoertner, Feb 11 2024

Extensions

More terms from Benoit Cloitre, Feb 03 2006

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

Views

Author

Gary Detlefs, Feb 20 2012

Keywords

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

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

A202156 y-values in the solution to x^2 - 13*y^2 = -1.

Original entry on oeis.org

5, 6485, 8417525, 10925940965, 14181862955045, 18408047189707445, 23893631070377308565, 31013914721302556809925, 40256037414619648361974085, 52252305550261582271285552405, 67823452348202119168480285047605, 88034788895660800419105138706238885
Offset: 1

Views

Author

Bruno Berselli, Dec 15 2011

Keywords

Comments

The corresponding values of x of this Pell equation are in A202155.

References

  • A. H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover Publications (New York), 1966, p. 264.

Links

Crossrefs

Cf. A002313, A003654, A006191, A031396, A075871, A202155.

Programs

  • Magma
    m:=13; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(5*x*(1-x)/(1-1298*x+x^2)));
  • Mathematica
    LinearRecurrence[{1298, -1}, {5, 6485}, 12]
  • Maxima
    makelist(expand(((18+5*sqrt(13))^(2*n-1)-(18-5*sqrt(13))^(2*n-1))/(2*sqrt(13))), n, 1, 12);

Formula

G.f.: 5*x*(1-x)/(1-1298*x+x^2).
a(n) = a(-n+1) = 5*(r^(2n-1)+1/r^(2n-1))/(r+1/r), where r=18+5*sqrt(13).
a(n) = A006191(6*n - 3). - Michael Somos, Feb 24 2023
Showing 1-3 of 3 results.

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