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%I #20 Sep 08 2022 08:45:48
%S 1,560,190399,64735100,22009743601,7483248089240,2544282340597999,
%T 865048512555230420,294113949986437744801,99997877946876278001920,
%U 33998984387987948082907999,11559554694037955471910717740,3930214596988516872501561123601
%N Subsequence of A167709 whose indices are congruent to 1 mod 5, i.e., a(n) = A167709(5*n+1).
%H G. C. Greubel, <a href="/A167822/b167822.txt">Table of n, a(n) for n = 0..100</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (340,-1).
%F a(n+2) = 340*a(n+1) - a(n).
%F a(n+1) = 170*a(n) + 39*sqrt(19*(a(n))^2 + 81).
%F G.f.: (1 + 220*z)/(1 - 340*z + z^2).
%F a(n) = (10*sqrt(19) + 19)/38*(170 + 39*sqrt(19))^n + (-10*sqrt(19) + 19)/38*(170 - 39*sqrt(19))^n.
%e a(0) = A167709(1) = 1, a(1) = A167709(6) = 560.
%p w(0):=1:for n from 0 to 20 do w(n+1):=170*w(n)+39*sqrt(19*(w(n))^2+81) :od: seq(w(n),n=0..20);for n from 0 to 20 do u(n):=simplify((10*sqrt(19)+19)/38*(170+39*sqrt(19))^(n)+(-10*sqrt(19)+19)/38*(170-39*sqrt(19))^(n)):od:seq(u(n),n=0..20);taylor(((1+560*z-1*340*z)/(1-340*z+z^2)),z=0,21);
%t LinearRecurrence[{340,-1},{1,560},50] (* _G. C. Greubel_, Jun 27 2016 *)
%t RecurrenceTable[{a[1] == 1, a[2] == 560, a[n] == 340 a[n-1] - a[n-2]}, a, {n, 15}] (* _Vincenzo Librandi_, Jun 28 2016 *)
%o (Magma) I:=[1,560]; [n le 2 select I[n] else 340*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Jun 28 2016
%K easy,nonn
%O 0,2
%A _Richard Choulet_, Nov 13 2009
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