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Search: id:A159548
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| A159548 |
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Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x. |
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4
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| 181, 199, 221, 865, 995, 1145, 5009, 5771, 6649, 29189, 33631, 38749, 170125, 196015, 225845, 991561, 1142459, 1316321, 5779241, 6658739, 7672081, 33683885, 38809975, 44716165, 196324069, 226201111, 260624909, 1144260529, 1318396691
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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(-19,a(1)) and (A129993(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+199)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 1.
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FORMULA
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a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=181, a(2)=199, a(3)=221, a(4)=865, a(5)=995, a(6)=1145.
G.f.: (1-x)*(181+380*x+601*x^2+380*x^3+181*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 199*A001653(k) for k >= 1.
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EXAMPLE
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(-19, a(1)) = (-19, 181) is a solution: (-19)^2+(-19+199)^2 = 361+32400 = 32761 = 181^2.
(A129993(1), a(2)) = (0, 199) is a solution: 0^2+(0+199)^2 = 39601 = 199^2.
(A129993(3), a(4)) = (504, 865) is a solution: 504^2+(504+199)^2 = 254016+494209 = 748225 = 865^2.
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PROGRAM
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(PARI) {forstep(n=-20, 50000000, [1, 3], if(issquare(2*n^2+398*n+39601, &k), print1(k, ", ")))}
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CROSSREFS
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Cf. A129993, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).
Sequence in context: A157747 A139649 A159263 this_sequence A139648 A142519 A053140
Adjacent sequences: A159545 A159546 A159547 this_sequence A159549 A159550 A159551
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KEYWORD
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nonn
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AUTHOR
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Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 14 2009
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