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%I #12 Apr 18 2024 12:51:33
%S 169,289,625,2809,7225,18769,93025,243049,635209,3157729,8254129,
%T 21576025,107267449,280395025,732947329,3643933225,9525174409,
%U 24898630849,123786459889,323575532569,845820499225,4205095700689
%N Squares of the form k^2+(k+17)^2 with integer k.
%C Square roots of k^2+(k+17)^2 are in A155923, values k (except for -5) are in A118120.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,34,-34,0,-1,1).
%F a(n) = 34*a(n-3)-a(n-6)-2312 for n > 6; a(1)=169, a(2)=289, a(3)=625, a(4)=2809, a(5)=7225, a(6)=18769.
%F G.f.: x*(169+120*x+336*x^2-3562*x^3+336*x^4+120*x^5+169*x^6)/((1-x)*(1-34*x^3+x^6)).
%F Limit_{n -> oo} a(n)/a(n-3) = (17+12*sqrt(2)).
%F Limit_{n -> oo} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2 for n mod 3 = {0, 2}.
%F Limit_{n -> oo} a(n)/a(n-1) = ((387+182*sqrt(2))/17^2)^2 for n mod 3 = 1.
%e 625 = 25^2 is of the form k^2+(k+17)^2 with k = 7: 7^2+24^2 = 625. Hence 625 is in the sequence.
%t LinearRecurrence[{1,0,34,-34,0,-1,1},{169,289,625,2809,7225,18769,93025},30] (* _Harvey P. Dale_, Apr 22 2022 *)
%o (PARI) {forstep(n=-5, 1600000, [1, 3], if(issquare(a=2*n*(n+17)+289), print1(a, ",")))}
%Y Equals A155923^2. Cf. A156160 (first trisection), A156161 (second trisection), A156162 (third trisection).
%Y Cf. A118120, A156035 (decimal expansion of 3+2*sqrt(2)), A156164 (decimal expansion of 17+12*sqrt(2)), A156163 (decimal expansion of (19+6*sqrt(2))/17), A157649 (decimal expansion of (387+182*sqrt(2))/17^2).
%K nonn,easy,changed
%O 1,1
%A _Klaus Brockhaus_, Feb 09 2009
%E G.f. corrected, fourth comment and cross-references edited by _Klaus Brockhaus_, Sep 23 2009
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