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A160209 Positive numbers y such that y^2 is of the form x^2+(x+937)^2 with integer x. 3
673, 937, 1685, 2353, 4685, 9437, 13445, 27173, 54937, 78317, 158353, 320185, 456457, 922945, 1866173, 2660425, 5379317, 10876853, 15506093, 31352957, 63394945, 90376133, 182738425, 369492817, 526750705, 1065077593, 2153561957 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-385, a(1)) and (A129974(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+937)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (1179+506*sqrt(2))/937 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (933747+224782*sqrt(2))/937^2 for n mod 3 = 1.

LINKS

Table of n, a(n) for n=1..27.

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)=673, a(2)=937, a(3)=1685, a(4)=2353, a(5)=4685, a(6)=9437.

G.f.: (1-x)*(673+1610*x+3295*x^2+1610*x^3+673*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 937*A001653(k) for k >= 1.

EXAMPLE

(-385, a(1)) = (-385, 673) is a solution: (-385)^2+(-385+937)^2 = 148225+304704 = 452929 = 673^2.

(A129974(1), a(2)) = (0, 937) is a solution: 0^2+(0+937)^2 = 877969 = 937^2.

(A129974(3), a(4)) = (1128, 2353) is a solution: 1128^2+(1128+937)^2 = 1272384+4264225 = 5536609 = 2353^2.

MATHEMATICA

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {673, 937, 1685, 2353, 4685, 9437}, 30] (* Harvey P. Dale, Dec 25 2017 *)

PROG

(PARI) {forstep(n=-388, 10000000, [3, 1], if(issquare(2*n^2+1874*n+877969, &k), print1(k, ", ")))}

CROSSREFS

Cf. A129974, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160210 (decimal expansion of (1179+506*sqrt(2))/937), A160211 (decimal expansion of (933747+224782*sqrt(2))/937^2).

Sequence in context: A047728 A297123 A335254 * A234117 A171266 A267818

Adjacent sequences:  A160206 A160207 A160208 * A160210 A160211 A160212

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, May 18 2009

STATUS

approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)