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%I #25 Oct 21 2022 22:09:02
%S 0,7,28,51,88,207,340,555,1248,2023,3276,7315,11832,19135,42676,69003,
%T 111568,248775,402220,650307,1450008,2344351,3790308,8451307,13663920,
%U 22091575,49257868,79639203,128759176,287095935,464171332,750463515,1673317776
%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.
%C Also values x of Pythagorean triples (x, x+17, y).
%C Corresponding values y of solutions (x, y) are in A155923.
%C For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a prime number in A066436, m >= 2 the associated value in A066049, the x values are given by the sequence defined by a(n) = 6*a(n-3)-a(n-6)+2p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21 (cf. A118673).
%C For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(0)=p, b(1)=2*m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. - _Mohamed Bouhamida_, Sep 09 2009
%H G. C. Greubel, <a href="/A118120/b118120.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F a(n) = 6*a(n-3) -a(n-6) +34 for n > 5; a(0)=0, a(1)=7, a(2)=28, a(3)=51, a(4)=88, a(5)=207.
%F G. f.: x*(7 +21*x +23*x^2 -5*x^3 -7*x^4 -5*x^5)/((1-x)*(1-6*x^3+x^6)).
%t Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+17)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,7,28,51,88,207,340},50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%o (Magma) [ n: n in [0..25000000] | IsSquare(2*n*(n+17)+289) ];
%o (PARI) m=32; v=concat([0, 7, 28, 51, 88, 207], vector(m-6)); for(n=7, m, v[n]=6*v[n-3]-v[n-6]+34); v
%Y Cf. A155923, A118673, A066436 (primes of the form 2*n^2-1), A066049 (2*n^2-1 is prime), A118554, A118611, A118630.
%Y Cf. A155464 (first trisection), A155465 (second trisection), A155466 (third trisection).
%K nonn,easy
%O 0,2
%A _Mohamed Bouhamida_, May 12 2006
%E Edited and 248755 changed to 248775 by _Klaus Brockhaus_, Feb 01 2009
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