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A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x. 3
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 (list; graph; refs; listen; history; text; internal format)
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.

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

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

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

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.: (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.

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

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Last modified October 22 10:24 EDT 2019. Contains 328317 sequences. (Running on oeis4.)