|
%I #11 Feb 15 2020 10:52:27
%S 0,132,2295,2859,3535,15792,19060,22984,94363,113407,136275,552292,
%T 663288,796572,3221295,3868227,4645063,18777384,22547980,27075712,
%U 109444915,131421559,157811115,637894012,765983280,919792884,3717921063
%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.
%C Also values x of Pythagorean triples (x, x+953, y).
%C Corresponding values y of solutions (x, y) are in A160212.
%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {1, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 0.
%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)+1906 for n > 6; a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792.
%F G.f.: x*(132+2163*x+564*x^2-116*x^3-721*x^4-116*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F a(3*k+1) = 953*A001652(k) for k >= 0.
%F a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792, a(7)=19060, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - _Harvey P. Dale_, Apr 12 2013
%t LinearRecurrence[{1,0,6,-6,0,-1,1},{0,132,2295,2859,3535,15792,19060},30] (* _Harvey P. Dale_, Apr 12 2013 *)
%o (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209), print1(n, ",")))}
%Y Cf. A160212, A001652, A129974, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).
%K nonn,easy
%O 1,2
%A _Mohamed Bouhamida_, Jun 13 2007
%E Edited and two terms added by _Klaus Brockhaus_, May 18 2009
|
