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 A157014 #46 May 09 2025 00:51:50
%S A157014 1,21,461,10121,222201,4878301,107100421,2351330961,51622180721,
%T A157014 1133336644901,24881784007101,546265911511321,11992968269241961,
%U A157014 263299036011811821,5780585823990618101,126909589091781786401,2786230374195208682721,61170158643202809233461
%N A157014 Expansion of x*(1-x)/(1 - 22*x + x^2).
%C A157014 This sequence is part of a solution of a general problem involving 2 equations, three sequences a(n), b(n), c(n) and a constant A:
%C A157014 A * c(n)+1 = a(n)^2,
%C A157014 (A+1) * c(n)+1 = b(n)^2, where solutions are given by the recurrences:
%C A157014 a(1) = 1, a(2) = 4*A+1, a(n) = (4*A+2)*a(n-1)-a(n-2) for n>2, resulting in a(n) terms 1, 4*A+1, 16*A^2+12*A+1, 64*A^3+80*A^2+24*A+1, ...;
%C A157014 b(1) = 1, b(2) = 4*A+3, b(n) = (4*A+2)*b(n-1)-b(n-2) for n>2, resulting in b(n) terms 1, 4*A+3, 16*A^2+20*A+5, 64*A^3+112*A^2+56*A+7, ...;
%C A157014 c(1) = 0, c(2) = 16*A+8, c(3) = (16*A^2+16*A+3)*c(2), c(n) = (16*A^2+16*A+3) * (c(n-1)-c(n-2)) + c(n-3) for n>3, resulting in c(n) terms 0, 16*A+8, 256*A^3+384*A^2+176*A+24, 4096*A^5 + 10240*A^4 + 9472*A^3 + 3968*A^2 + 736*A + 48, ... .
%C A157014 A157014 is the a(n) sequence for A=5.
%C A157014 For other A values the a(n), b(n) and c(n) sequences are in the OEIS:
%C A157014 A a-sequence b-sequence c-sequence
%C A157014 1 A001653 A002315(n-1) A078522
%C A157014 2 A072256 A054320(n-1) A045502(n-1)
%C A157014 3 A001570 A028230 A059989(n-1)
%C A157014 4 A007805 A049629(n-1) A157459
%C A157014 5 -> A157014 <- A133283 A157460
%C A157014 6 A153111 A157461 A157874
%C A157014 7 A157877 A157878 A157879
%C A157014 8 A077420(n-1) A046176 A157880
%C A157014 9 A097315(n-1) A097314(n-1) A157881
%C A157014 Positive values of x (or y) satisfying x^2 - 22xy + y^2 + 20 = 0. - _Colin Barker_, Feb 19 2014
%C A157014 From _Klaus Purath_, Apr 22 2025: (Start)
%C A157014 Nonnegative solutions to the Diophantine equation 5*b(n)^2 - 6*a(n)^2 = -1. The corresponding b(n) are A133283(n). Note that (b(n+1)^2 - b(n)*b(n+2))/4 = 6 and (a(n)*a(n+2) - a(n+1)^2)/4 = 5.
%C A157014 (a(n) + b(n))/2 = (b(n+1) - a(n+1))/2 = A077421(n-1) = Lucas U(22,1). Also b(n)*a(n+1) - b(n+1)*a(n) = -2.
%C A157014 a(n)=(t(i+2*n-1) + t(i))/(t(i+n) + t(i+n-1)) as long as t(i+n) + t(i+n-1) != 0 for any integer i and n >= 1 where (t) is a sequence satisfying t(i+3) = 21*t(i+2) - 21*t(i+1) + t(i) or t(i+2) = 22*t(i+1) - t(i) without regard to initial values and including this sequence itself. (End)
%H A157014 Vincenzo Librandi, <a href="/A157014/b157014.txt">Table of n, a(n) for n = 1..200</a>
%H A157014 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (22,-1).
%F A157014 G.f.: x*(1-x)/(1-22*x+x^2).
%F A157014 a(1) = 1, a(2) = 21, a(n) = 22*a(n-1) - a(n-2) for n>2.
%F A157014 5*A157460(n)+1 = a(n)^2 for n>=1.
%F A157014 6*A157460(n)+1 = A133283(n)^2 for n>=1.
%F A157014 a(n) = (6+sqrt(30)-(-6+sqrt(30))*(11+2*sqrt(30))^(2*n))/(12*(11+2*sqrt(30))^n). - _Gerry Martens_, Jun 07 2015
%F A157014 a(n) = ChebyshevU(n-1, 11) - ChebyshevU(n-2, 11). - _G. C. Greubel_, Jan 14 2020
%p A157014 seq( simplify(ChebyshevU(n-1,11) - ChebyshevU(n-2,11)), n=1..20); # _G. C. Greubel_, Jan 14 2020
%t A157014 CoefficientList[Series[(1-x)/(1-22x+x^2), {x,0,20}], x] (* _Vincenzo Librandi_, Feb 21 2014 *)
%t A157014 a[c_, n_] := Module[{},
%t A157014 p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
%t A157014 d := Denominator[Convergents[Sqrt[c], n p]];
%t A157014 t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
%t A157014 Return[t];
%t A157014 ] (* Complement of A041049 *)
%t A157014 a[30, 20] (* _Gerry Martens_, Jun 07 2015 *)
%t A157014 Table[ChebyshevU[n-1, 11] - ChebyshevU[n-2, 11], {n,20}] (* _G. C. Greubel_, Jan 14 2020 *)
%o A157014 (PARI) Vec((1-x)/(1-22*x+x^2)+O(x^20)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A157014 (Magma) I:=[1,21]; [n le 2 select I[n] else 22*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Feb 21 2014
%o A157014 (Sage) [chebyshev_U(n-1,11) - chebyshev_U(n-2,11) for n in (1..20)] # _G. C. Greubel_, Jan 14 2020
%o A157014 (GAP) a:=[1,21];; for n in [3..20] do a[n]:=22*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, Jan 14 2020
%Y A157014 Cf. similar sequences listed in A238379.
%K A157014 nonn,easy
%O A157014 1,2
%A A157014 _Paul Weisenhorn_, Feb 21 2009
%E A157014 Edited by _Alois P. Heinz_, Sep 09 2011