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A130004 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2. 6
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; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..30.

Index to sequences with linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

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.

MATHEMATICA

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 31, 1204, 1347, 1504, 8151, 8980}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)

PROG

(PARI) {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+898*n+201601), print1(n, ", ")))}

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

KEYWORD

nonn,easy

AUTHOR

Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Jun 15 2007

EXTENSIONS

Edited and two terms added by Klaus Brockhaus, Apr 17 2009

STATUS

approved

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Last modified April 20 16:25 EDT 2014. Contains 240807 sequences.