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a(0)=1, a(1)=193; for n>1, a(n) = 194*a(n-1) - a(n-2).
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%I #26 Jul 09 2022 21:47:43

%S 1,193,37441,7263361,1409054593,273349327681,53028360515521,

%T 10287228590683393,1995669318232062721,387149560508429484481,

%U 75105019069317087926593,14569986549887006628274561,2826502285659009968797338241,548326873431298046940055344193,106372586943386162096401939435201

%N a(0)=1, a(1)=193; for n>1, a(n) = 194*a(n-1) - a(n-2).

%C Let G and H be sequences of the form G(i) = 4*G(i-1) - G(i-2) and H(j) = 14*H(j-1) - H(j-2) for arbitrary integers i, j and without regard to initial values of G or H, then a(n) = (G(i) + G(i+8*n+4))/(14*G(i+4*n+2)) = (H(j) + H(j+4*n+2))/(14*H(j+2*n+1)) with the exception of G(i+4*n+2) or H(j+2*n+1) != 0. - _Klaus Purath_, Aug 31 2020

%H Colin Barker, <a href="/A302332/b302332.txt">Table of n, a(n) for n = 0..400</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (194,-1).

%F G.f.: (1 - x)/(1 - 194*x + x^2).

%F a(n) = a(-1-n).

%F a(n) = cosh((2*n + 1)*arccosh(7))/7.

%F a(n) = ((7 + 4*sqrt(3))^(2*n + 1) + 1/(7 + 4*sqrt(3))^(2*n + 1))/14.

%F a(n) = (a(n-1)^2 + 192)/a(n-2). - _Klaus Purath_, Aug 31 2020

%F a(n) = (1/7)*T(2*n+1, 7), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Jul 08 2022

%t LinearRecurrence[{194, -1}, {1, 193}, 20]

%o (PARI) x='x+O('x^99); Vec((1-x)/(1-194*x+x^2)) \\ _Altug Alkan_, Apr 06 2018

%Y Seventh row of the array A188646.

%Y First bisection of A041269, A042127.

%Y Similar sequences of the type cosh((2*n+1)*arccosh(k))/k are listed in A302329.

%K nonn,easy

%O 0,2

%A _Bruno Berselli_, Apr 05 2018