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A157257
Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.
6
29, 41, 85, 89, 205, 481, 505, 1189, 2801, 2941, 6929, 16325, 17141, 40385, 95149, 99905, 235381, 554569, 582289, 1371901, 3232265, 3393829, 7996025, 18839021, 19780685, 46604249, 109801861, 115290281, 271629469, 639972145, 671961001
OFFSET
1,1
COMMENTS
(-20, a(1)) and (A129288(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+41)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(7-2*sqrt(2))^2/(7+2*sqrt(2))^2 for n mod 3 = 1.
FORMULA
a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=29, a(2)=41, a(3)=85, a(4)=89, a(5)=205, a(6)=481.
G.f.: (1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 41*A001653(k) for k >= 1.
EXAMPLE
(-20, a(1)) = (-20, 29) is a solution: (-20)^2+(-20+41)^2 = 400+441 = 841 = 29^2.
(A129288(1), a(2)) = (0, 41) is a solution: 0^2+(0+41)^2 = 1681 = 41^2.
(A129288(3), a(4)) = (39, 89) is a solution: 39^2+(39+41)^2 = 1521+6400 = 7921 = 89^2.
MATHEMATICA
CoefficientList[Series[(1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6), {x, 0, 50}], x] (* G. C. Greubel, Feb 04 2018 *)
PROG
(PARI) {forstep(n=-20, 500000000, [3 , 1], if(issquare(n^2+(n+41)^2, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6)) \\ G. C. Greubel, Feb 04 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6))) // G. C. Greubel, Feb 04 2018
CROSSREFS
Cf. A129288, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).
Sequence in context: A058900 A137226 A057539 * A214405 A104072 A357175
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Feb 26 2009
STATUS
approved