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

0, 2145, 3773, 6468, 8540, 12005, 19208, 24521, 28665, 35672, 41148, 50421, 61388, 69972, 84525, 95921, 115248, 156065, 186480, 210308, 250733, 282405, 336140, 399797, 449673, 534296, 600600, 713097, 950796, 1127973, 1266797, 1502340

Offset: 1

Author

Mohamed Bouhamida, May 16 2006

Keywords

• nonn
• easy

Comments

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

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

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

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}.

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}.

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

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,0,0,-1,1).

Crossrefs

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

Programs

PARI

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

Formula

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.

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)).

Extensions

Edited by Klaus Brockhaus, Feb 14 2009