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A130004
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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.
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6
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0, 31, 1204, 1347, 1504, 8151, 8980, 9891, 48600, 53431, 58740, 284347, 312504, 343447, 1658380, 1822491, 2002840, 9666831, 10623340, 11674491, 56343504, 61918447, 68045004, 328395091, 360888240, 396596431, 1914027940, 2103411891, 2311534480, 11155773447
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also values x of Pythagorean triples (x, x+449, y).
Corresponding values y of solutions (x, y) are in A159589.
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) = (451+30*sqrt(2))/449 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 0.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).
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FORMULA
| a(n) = 6*a(n-3)-a(n-6)+898 for n > 6; a(1)=0, a(2)=31, a(3)=1204, a(4)=1347, a(5)=1504, a(6)=8151.
G.f.: x*(31+1173*x+143*x^2-29*x^3-391*x^4-29*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 449*A001652(k) for k >= 0.
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MATHEMATICA
| LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 31, 1204, 1347, 1504, 8151, 8980}, 50] (* From Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)
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PROG
| (PARI) {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+898*n+201601), print1(n, ", ")))}
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CROSSREFS
| Cf. A159589, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).
Sequence in context: A015257 A199234 A123826 * A182784 A139162 A078961
Adjacent sequences: A130001 A130002 A130003 * A130005 A130006 A130007
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Jun 15 2007
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EXTENSIONS
| Edited and two terms added by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 17 2009
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