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%I #41 Sep 08 2022 08:45:41
%S 1,21,461,10121,222201,4878301,107100421,2351330961,51622180721,
%T 1133336644901,24881784007101,546265911511321,11992968269241961,
%U 263299036011811821,5780585823990618101,126909589091781786401,2786230374195208682721,61170158643202809233461
%N Expansion of x*(1-x)/(1 - 22*x + x^2).
%C 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 A * c(n)+1 = a(n)^2,
%C (A+1) * c(n)+1 = b(n)^2, where solutions are given by the recurrences:
%C 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 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 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 is the a(n) sequence for A=5.
%C For other A values the a(n), b(n) and c(n) sequences are in the OEIS:
%C A a-sequence b-sequence c-sequence
%C 1 A001653 A002315(n-1) A078522
%C 2 A072256 A054320(n-1) A045502(n-1)
%C 3 A001570 A028230 A059989(n-1)
%C 4 A007805 A049629(n-1) A157459
%C 5 -> A157014 <- A133283 A157460
%C 6 A153111 A157461 A157874
%C 7 A157877 A157878 A157879
%C 8 A077420(n-1) A046176 A157880
%C 9 A097315(n-1) A097314(n-1) A157881
%C Positive values of x (or y) satisfying x^2 - 22xy + y^2 + 20 = 0. - _Colin Barker_, Feb 19 2014
%H Vincenzo Librandi, <a href="/A157014/b157014.txt">Table of n, a(n) for n = 1..200</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (22,-1).
%F G.f.: x*(1-x)/(1-22*x+x^2).
%F a(1) = 1, a(2) = 21, a(n) = 22*a(n-1) - a(n-2) for n>2.
%F 5*A157460(n)+1 = a(n)^2 for n>=1.
%F 6*A157460(n)+1 = A133283(n)^2 for n>=1.
%F 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 a(n) = ChebyshevU(n-1, 11) - ChebyshevU(n-2, 11). - _G. C. Greubel_, Jan 14 2020
%p seq( simplify(ChebyshevU(n-1,11) - ChebyshevU(n-2,11)), n=1..20); # _G. C. Greubel_, Jan 14 2020
%t CoefficientList[Series[(1-x)/(1-22x+x^2), {x,0,20}], x] (* _Vincenzo Librandi_, Feb 21 2014 *)
%t a[c_, n_] := Module[{},
%t p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
%t d := Denominator[Convergents[Sqrt[c], n p]];
%t t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
%t Return[t];
%t ] (* Complement of A041049 *)
%t a[30, 20] (* _Gerry Martens_, Jun 07 2015 *)
%t Table[ChebyshevU[n-1, 11] - ChebyshevU[n-2, 11], {n,20}] (* _G. C. Greubel_, Jan 14 2020 *)
%o (PARI) Vec((1-x)/(1-22*x+x^2)+O(x^20)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (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 (Sage) [chebyshev_U(n-1,11) - chebyshev_U(n-2,11) for n in (1..20)] # _G. C. Greubel_, Jan 14 2020
%o (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 Cf. similar sequences listed in A238379.
%K nonn,easy
%O 1,2
%A _Paul Weisenhorn_, Feb 21 2009
%E Edited by _Alois P. Heinz_, Sep 09 2011
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