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%I A118611 #18 Mar 16 2024 15:21:20
%S A118611 0,77,132,245,392,585,728,1029,1428,1725,2352,3185,4292,5117,6860,
%T A118611 9177,10904,14553,19404,25853,30660,40817,54320,64385,85652,113925,
%U A118611 151512,179529,238728,317429,376092,500045,664832,883905,1047200,1392237,1850940,2192853
%N A118611 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.
%C A118611 Also values x of Pythagorean triples (x, x+343, y); 343=7^3.
%C A118611 Corresponding values y of solutions (x, y) are in A157246.
%C A118611 Limit_{n -> oo} a(n)/a(n-7) = 3+2*sqrt(2).
%C A118611 Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 7 = {1, 2, 4, 5, 6}.
%C A118611 Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 7 = {0, 3}.
%H A118611 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,6,-6,0,0,0,0,0,-1,1).
%F A118611 a(n) = 6*a(n-7)-a(n-14)+686 for n > 14; a(1)=0, a(2)=77, a(3)=132, a(4)=245, a(5)=392, a(6)=585, a(7)=728, a(8)=1029, a(9)=1428, a(10)=1725, a(11)=2352, a(12)=3185, a(13)=4292, a(14)=5117.
%F A118611 G.f.: x*(77+55*x+113*x^2+147*x^3+193*x^4+143*x^5+301*x^6-63*x^7 -33*x^8-51*x^9-49*x^10-51*x^11-33*x^12-63*x^13)/((1-x)*(1-6*x^7+x^14)).
%F A118611 a(7*k+1) = 343*A001652(k) for k >= 0.
%e A118611 132^2+(132+343)^2 = 17424+225625 = 243049 = 493^2.
%t A118611 LinearRecurrence[{1, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, -1, 1}, {0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 13 2012 *)
%o A118611 (PARI) {forstep(n=0, 1400000, [1, 3], if(issquare(n^2+(n+343)^2), print1(n, ",")))}
%Y A118611 Cf. A157246, A001652, A118576, A118554, A118611, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).
%K A118611 nonn,easy
%O A118611 1,2
%A A118611 _Mohamed Bouhamida_, May 08 2006
%E A118611 Edited by _Klaus Brockhaus_, Feb 25 2009

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