A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x.

181, 199, 221, 865, 995, 1145, 5009, 5771, 6649, 29189, 33631, 38749, 170125, 196015, 225845, 991561, 1142459, 1316321, 5779241, 6658739, 7672081, 33683885, 38809975, 44716165, 196324069, 226201111, 260624909, 1144260529, 1318396691

Offset

1,1

Comments

(-19,a(1)) and (A129993(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+199)^2 = y^2.

Links

Table of n, a(n) for n=1..29.

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

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=181, a(2)=199, a(3)=221, a(4)=865, a(5)=995, a(6)=1145.

G.f.: x*(1-x)*(181+380*x+601*x^2+380*x^3+181*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 199*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) = (201+20*sqrt(2))/199 for n mod 3 = {0, 2}.

Limit_{n -> oo} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 1.

Example

(-19, a(1)) = (-19, 181) is a solution: (-19)^2+(-19+199)^2 = 361+32400 = 32761 = 181^2.
(A129993(1), a(2)) = (0, 199) is a solution: 0^2+(0+199)^2 = 39601 = 199^2.
(A129993(3), a(4)) = (504, 865) is a solution: 504^2+(504+199)^2 = 254016+494209 = 748225 = 865^2.

Prog

(PARI) {forstep(n=-20, 50000000, [1, 3], if(issquare(2*n^2+398*n+39601, &k), print1(k, ", ")))}

Crossrefs

Cf. A129993, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

Sequence in context: A253156 A139649 A159263 * A206281 A224614 A216309

Adjacent sequences: A159545 A159546 A159547 * A159549 A159550 A159551

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 14 2009

Status

approved