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 A097780 #42 Sep 02 2021 04:01:11
%S A097780 1,25,624,15575,388751,9703200,242191249,6045078025,150884759376,
%T A097780 3766073906375,94000962899999,2346257998593600,58562449001940001,
%U A097780 1461714967049906425,36484311727245720624,910646078214093109175
%N A097780 Chebyshev polynomials S(n,25) with Diophantine property.
%C A097780 All positive integer solutions of Pell equation b(n)^2 - 621*a(n)^2 = +4 together with b(n)=A090733(n+1), n>=0. Note that D=621=69*3^2 is not squarefree.
%C A097780 For positive n, a(n) equals the permanent of the tridiagonal matrix with 25's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011
%C A097780 For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,24}. - _Milan Janjic_, Jan 25 2015
%H A097780 Indranil Ghosh, <a href="/A097780/b097780.txt">Table of n, a(n) for n = 0..714</a>
%H A097780 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A097780 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A097780 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (25,-1).
%F A097780 a(n) = S(n, 25)=U(n, 25/2) = S(2*n+1, sqrt(25))/sqrt(25) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x).
%F A097780 a(n) = 25*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=25; a(-1)=0.
%F A097780 a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (25+3*sqrt(69))/2 and am := (25-3*sqrt(69))/2.
%F A097780 G.f.: 1/(1-25*x+x^2).
%F A097780 a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*24^k. - _Philippe Deléham_, Feb 10 2012
%F A097780 Product {n >= 0} (1 + 1/a(n)) = 1/23*(23 + 3*sqrt(69)). - _Peter Bala_, Dec 23 2012
%F A097780 Product {n >= 1} (1 - 1/a(n)) = 1/50*(23 + 3*sqrt(69)). - _Peter Bala_, Dec 23 2012
%e A097780 (x,y) = (2,0), (25;1), (623;25), (15550;624), ... give the nonnegative integer solutions to x^2 - 69*(3*y)^2 =+4.
%t A097780 LinearRecurrence[{25,-1},{1,25},20] (* _Harvey P. Dale_, Aug 23 2021 *)
%o A097780 (Sage) [lucas_number1(n,25,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%K A097780 nonn,easy
%O A097780 0,2
%A A097780 _Wolfdieter Lang_, Aug 31 2004