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.
%I A207832 #43 Oct 26 2024 03:32:04
%S A207832 0,2,36,646,11592,208010,3732588,66978574,1201881744,21566892818,
%T A207832 387002188980,6944472508822,124613502969816,2236098580947866,
%U A207832 40125160954091772,720016798592704030
%N A207832 Numbers x such that 20*x^2 + 1 is a perfect square.
%C A207832 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.
%C A207832 .
%C A207832 k x y
%C A207832 - ----------------------- -----------------------
%C A207832 2 A001542 {0,2,6,-1} A001541 {1,3,6,-1}
%C A207832 3 A001353 {0,1,4,-1} A001075 {1,2,4,-1}
%C A207832 5 A060645 {0,4,18,-1} A023039 {1,9,18,-1}
%C A207832 6 A001078 {0,2,10,-1} A001079 {1,5,10,-1}
%C A207832 7 A001080 {0,3,16,-1} A001081 {1,8,16,-1}
%C A207832 8 A001109 {0,1,6,-1} A001541 {1,3,6,-1}
%C A207832 10 A084070 {0,1,38,-1} A078986 {1,19,38,-1}
%C A207832 11 A001084 {0,3,20,-1} A001085 {1,10,20,-1}
%C A207832 12 A011944 {0,2,14,-1} A011943 {1,7,14,-1}
%C A207832 13 A075871 {0,180,1298,-1} A114047 {1,649,1298,-1}
%C A207832 14 A068204 {0,4,30,-1} A069203 {1,15,30,-1}
%C A207832 15 A001090 {0,1,8,-1} A001091 {1,4,8,-1}
%C A207832 17 A121740 {0,8,66,-1} A099370 {1,33,66,-1}
%C A207832 18 A202299 {0,4,34,-1} A056771 {1,17,34,-1}
%C A207832 19 A174765 {0,39,340,-1} A114048 {1,179,340,-1}
%C A207832 20 a(n) {0,2,18,-1} A023039 {1,9,18,-1}
%C A207832 21 A174745 {0,12,110,-1} A114049 {1,55,110,-1}
%C A207832 22 A174766 {0,42,394,-1} A114050 {1,197,394,-1}
%C A207832 23 A174767 {0,5,48,-1} A114051 {1,24,48,-1}
%C A207832 24 A004189 {0,1,10,-1} A001079 {1,5,10,-1}
%C A207832 26 A174768 {0,10,102,-1} A099397 {1,51,102,-1}
%C A207832 The sequence of the c parameter is listed in A180495.
%H A207832 Bruno Berselli, <a href="/A207832/b207832.txt">Table of n, a(n) for n = 0..500</a>
%H A207832 Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
%H A207832 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-1).
%F A207832 a(n) = 18*a(n-1) - a(n-2).
%F A207832 From _Bruno Berselli_, Feb 21 2012: (Start)
%F A207832 G.f.: 2*x/(1-18*x+x^2).
%F A207832 a(n) = -a(-n) = 2*A049660(n) = ((2 + sqrt(5))^(2*n)-(2 - sqrt(5))^(2*n))/(4*sqrt(5)). (End)
%F A207832 a(n) = Fibonacci(6*n)/4. - _Bruno Berselli_, Jun 19 2019
%F A207832 For n>=1, a(n) = A079962(6n-3). - _Christopher Hohl_, Aug 22 2021
%p A207832 readlib(issqr):for x from 1 to 720016798592704030 do if issqr(20*x^2+1) then print(x) fi od;
%t A207832 LinearRecurrence[{18, -1}, {0, 2}, 16] (* _Bruno Berselli_, Feb 21 2012 *)
%t A207832 Table[2 ChebyshevU[-1 + n, 9], {n, 0, 16}] (* _Herbert Kociemba_, Jun 05 2022 *)
%o A207832 (Magma) m:=16; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x/(1-18*x+x^2))); // _Bruno Berselli_, Jun 19 2019
%o A207832 (Maxima) makelist(expand(((2+sqrt(5))^(2*n)-(2-sqrt(5))^(2*n))/(4*sqrt(5))), n, 0, 15); /* _Bruno Berselli_, Jun 19 2019 */
%Y A207832 Cf. A023039, A049660, A079962.
%K A207832 nonn,easy
%O A207832 0,2
%A A207832 _Gary Detlefs_, Feb 20 2012