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URL: https://oeis.org/A129999

⇱ A129999 - OEIS

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A129999
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+337)^2 = y^2.
5
0, 27, 888, 1011, 1148, 6027, 6740, 7535, 35948, 40103, 44736, 210335, 234552, 261555, 1226736, 1367883, 1525268, 7150755, 7973420, 8890727, 41678468, 46473311, 51819768, 242920727, 270867120, 302028555, 1415846568, 1578730083
OFFSET
1,2
COMMENTS
Also values x of Pythagorean triples (x, x+337, y).
Corresponding values y of solutions (x, y) are in A159574.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (339+26*sqrt(2))/337 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (278307+179662*sqrt(2))/337^2 for n mod 3 = 0.
FORMULA
a(n)=6*a(n-3)-a(n-6)+674 for n > 6; a(1)=0, a(2)=27, a(3)=888, a(4)=1011, a(5)=1148, a(6)=6027.
G.f.: x*(27+861*x+123*x^2-25*x^3-287*x^4-25*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 337*A001652(k) for k >= 0.
a(0)=0, a(1)=27, a(2)=888, a(3)=1011, a(4)=1148, a(5)=6027, a(6)=6740, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Feb 26 2015
MATHEMATICA
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 27, 888, 1011, 1148, 6027, 6740}, 40] (* Harvey P. Dale, Feb 26 2015 *)
PROG
(PARI) {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+674*n+113569), print1(n, ", ")))}
CROSSREFS
Cf. A159574, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159575 (decimal expansion of (339+26*sqrt(2))/337), A159576 (decimal expansion of (278307+179662*sqrt(2))/337^2).
Sequence in context: A376778 A061695 A107050 * A132059 A292362 A239571
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, Jun 15 2007
EXTENSIONS
Edited and two terms added by Klaus Brockhaus, Apr 16 2009
STATUS
approved