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Search: id:A159589
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| A159589 |
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Positive numbers y such that y^2 is of the form x^2+(x+449)^2 with integer x. |
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4
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| 421, 449, 481, 2045, 2245, 2465, 11849, 13021, 14309, 69049, 75881, 83389, 402445, 442265, 486025, 2345621, 2577709, 2832761, 13671281, 15023989, 16510541, 79682065, 87566225, 96230485, 464421109, 510373361, 560872369, 2706844589
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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(-29,a(1)) and (A130004(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.
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 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 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)=421, a(2)=449, a(3)=481, a(4)=2045, a(5)=2245, a(6)=2465.
G.f.: (1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 449*A001653(k) for k >= 1.
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EXAMPLE
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(-29, a(1)) = (-29, 421) is a solution: (-29)^2+(-29+449)^2 = 841+176400 = 177241 = 421^2.
(A130004(1), a(2)) = (0, 449) is a solution: 0^2+(0+449)^2 = 201601 = 449^2.
(A130004(3), a(4)) = (1204, 2045) is a solution: 1204^2+(1204+449)^2 = 1449616+2732409 = 4182025 = 2045^2.
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PROGRAM
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(PARI) {forstep(n=-32, 50000000, [3, 1], if(issquare(2*n^2+898*n+201601, &k), print1(k, ", ")))}
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CROSSREFS
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Cf. A130004, A001653, 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: A059951 A028683 A014890 this_sequence A031784 A051648 A146358
Adjacent sequences: A159586 A159587 A159588 this_sequence A159590 A159591 A159592
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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 18 2009
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