A159626 Positive numbers y such that y^2 is of the form x^2+(x+577)^2 with integer x.

545, 577, 613, 2657, 2885, 3133, 15397, 16733, 18185, 89725, 97513, 105977, 522953, 568345, 617677, 3047993, 3312557, 3600085, 17765005, 19306997, 20982833, 103542037, 112529425, 122296913, 603487217, 655869553, 712798645, 3517381265

Offset: 1

Author

Klaus Brockhaus, Apr 21 2009

Keywords

• nonn
• easy

Comments

(-33,a(1)) and (A130005(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+577)^2 = y^2.

Examples

			(-33, a(1)) = (-33, 545) is a solution: (-33)^2+(-33+577)^2 = 1089+295936 = 297025 = 545^2.
(A130005(1), a(2)) = (0, 577) is a solution: 0^2+(0+577)^2 = 332929 = 577^2.
(A130005(3), a(4)) = (1568, 2657) is a solution: 1568^2+(1568+577)^2 = 2458624+4601025 = 7059649 = 2657^2.
		

Links

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

Crossrefs

Cf. A130005, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159627 (decimal expansion of (579+34*sqrt(2))/577), A159628 (decimal expansion of (855171+556990*sqrt(2))/577^2).

Programs

PARI

{forstep(n=-36, 50000000, [3, 1], if(issquare(2*n^2+1154*n+332929, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=545, a(2)=577, a(3)=613, a(4)=2657, a(5)=2885, a(6)=3133.

G.f.: (1-x)*(545+1122*x+1735*x^2+1122*x^3+545*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 577*A001653(k) for k >= 1.

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

Limit_{n -> oo} a(n)/a(n-1) = (579+34*sqrt(2))/577 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (855171+556990*sqrt(2))/577^2 for n mod 3 = 1.