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A118576 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+16807)^2 = y^2. 4

%I #17 Mar 16 2024 14:24:29

%S 0,2145,3773,6468,8540,12005,19208,24521,28665,35672,41148,50421,

%T 61388,69972,84525,95921,115248,156065,186480,210308,250733,282405,

%U 336140,399797,449673,534296,600600,713097,950796,1127973,1266797,1502340

%N Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+16807)^2 = y^2.

%C Also values x of Pythagorean triples (x, x+16807, y); 16807 = 7^5.

%C Corresponding values y of solutions (x, y) are in A156713.

%C Limit_{n -> oo} a(n)/a(n-11) = 3+2*sqrt(2).

%C Limit_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 11 = {1, 2, 4, 6, 8, 10}.

%C Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 11 = {0, 3, 5, 9}.

%C Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 11 = 7.

%H <a href="/index/Rec#order_23">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,0,0,-1,1).

%F a(n) = 6*a(n-11)-a(n-22)+33614 for n > 22; a(1) = 0, a(2) = 2145, a(3) = 3773, a(4) = 6468, a(5) = 8540, a(6) = 12005, a(7) = 19208, a(8) = 24521, a(9) = 28665, a(10) = 35672, a(11) = 41148, a(12) = 50421, a(13) = 61388, a(14) = 69972, a(15) = 84525, a(16) = 95921, a(17) = 115248, a(18) = 156065, a(19) = 186480, a(20) = 210308, a(21) = 250733, a(22) = 282405.

%F G.f.: x*(2145+1628*x+2695*x^2+2072*x^3+3465*x^4+7203*x^5+5313*x^6+4144*x^7+7007*x^8+5476*x^9+9273*x^10-1903*x^11-1184*x^12-1617*x^13-1036*x^14-1463*x^15-2401*x^16-1463*x^17 -1036*x^18-1617*x^19-1184*x^20-1903*x^21 )/((1-x)*(1-6*x^11+x^22)).

%o (PARI) {forstep(n=0, 1600000, [1, 3], if(issquare(2*n^2 + 33614*n + 282475249), print1(n, ",")))}

%Y Cf. A156713, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

%K nonn,easy,changed

%O 1,2

%A _Mohamed Bouhamida_, May 16 2006

%E Edited by _Klaus Brockhaus_, Feb 14 2009

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)