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%I #41 Jan 03 2019 17:46:53
%S 5,29,5741,33461,44560482149,1746860020068409,68480406462161287469,
%T 13558774610046711780701,4125636888562548868221559797461449,
%U 4760981394323203445293052612223893281
%N Prime hypotenuses of Pythagorean triangles with consecutive integer sides.
%C These primes belong to A001653.
%C From _Jianing Song_, Jan 02 2019: (Start)
%C Essentially the same sequence as A086383.
%C If p is a term then it is a unique-period prime in base sqrt(2*p^2 - 1). (End)
%H Robert Israel, <a href="/A056869/b056869.txt">Table of n, a(n) for n = 1..22</a>
%F a(n) = A086383(n+1). - _Jianing Song_, Jan 02 2019
%e 29 is included because it is prime and it is the hypotenuse of the 20, 21, 29 Pythagorean triangle.
%p f:= gfun:-rectoproc({a(n)=6*a(n-1)-a(n-2),a(1)=1,a(2)=5},a(n),remember):
%p select(isprime, [seq(f(n),n=1..1000)]); # _Robert Israel_, Oct 13 2015
%t Select[Sqrt[#^2+(#+1)^2]&/@With[{p=3+2Sqrt[2]},NestList[Floor[p #]+3&,3,120]],PrimeQ] (* _Harvey P. Dale_, May 02 2018 *)
%o (PARI) t(n) = if(n<3, 5^(n-1), 6*t(n-1)-t(n-2));
%o for(n=1, 50, if(isprime(t(n)), print1(t(n)", "))) \\ _Altug Alkan_, Oct 13 2015
%o (GAP) f:=[1,5];; for n in [3..60] do f[n]:=6*f[n-1]-f[n-2]; od; a:=Filtered(f,IsPrime);; Print(a); # _Muniru A Asiru_, Jan 03 2019
%Y Cf. A000129, A001653, A001652, A046090, A086383.
%K nonn
%O 1,1
%A _Harvey P. Dale_, Sep 02 2000
%E Incorrect link to index entries for linear recurrences with constant coefficients removed by _Colin Barker_, Oct 13 2015
%E Offset changed to 1 by _Colin Barker_, Oct 13 2015
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