A129837 - OEIS

%I #10 Feb 15 2020 10:52:27

%S 0,24,49,57,85,136,180,196,261,357,481,616,660,816,1105,1357,1449,

%T 1824,2380,3100,3885,4141,5049,6732,8200,8736,10921,14161,18357,22932,

%U 24424,29716,39525,48081,51205,63940,82824,107280,133945,142641,173485,230656

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+119)^2 = y^2.

%C Also values x of Pythagorean triples (x, x+119, y).

%C Corresponding values y of solutions (x, y) are in A156650.

%C lim_{n -> infinity} a(n)/a(n-9) = 3+2*sqrt(2).

%C lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)/((19+6*sqrt(2))/17) for n mod 9 = {1, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = ((19+6*sqrt(2))/17)^2/((9+4*sqrt(2))/7) for n mod 9 = {0, 3}.

%C lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 9 = {4, 8}.

%C lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 9 = {5, 7}.

%C lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((9+4*sqrt(2))/7)^2 for n mod 9 = 6.

%H <a href="/index/Rec#order_19">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 6, -6, 0, 0, 0, 0, 0, 0, 0, -1, 1).

%F a(n) = 6*a(n-9)-a(n-18)+238 for n > 18; a(1)=0, a(2)=24, a(3)=49, a(4)=57, a(5)=85, a(6)=136, a(7)=180, a(8)=196, a(9)=261, a(10)=357, a(11)=481, a(12)=616, a(13)=660, a(14)=816, a(15)=1105, a(16)=1357, a(17)=1449, a(18)=1824.

%F G.f.: x*(24+25*x+8*x^2+28*x^3+51*x^4+44*x^5+16*x^6+65*x^7+96*x^8-20*x^9-15*x^10-4*x^11-12*x^12-17*x^13-12*x^14-4*x^15-15*x^16-20*x^17 )/((1-x)*(1-6*x^9+x^18))

%t LinearRecurrence[{1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,24,49,57,85,136,180,196,261,357,481,616,660,816,1105,1357,1449,1824,2380}, 140] (* _Vladimir Joseph Stephan Orlovsky_, Feb 07 2012 *)

%o (PARI) {forstep(n=0, 240000, [1, 3], if(issquare(n^2+(n+119)^2), print1(n, ",")))}

%Y Cf. A156650, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17), A118630.

%K nonn

%O 1,2

%A _Mohamed Bouhamida_, May 21 2007

%E Edited and extended by _Klaus Brockhaus_, Feb 13 2009