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Search: id:A159641
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| A159641 |
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Positive numbers y such that y^2 is of the form x^2+(x+647)^2 with integer x. |
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4
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| 613, 647, 685, 2993, 3235, 3497, 17345, 18763, 20297, 101077, 109343, 118285, 589117, 637295, 689413, 3433625, 3714427, 4018193, 20012633, 21649267, 23419745, 116642173, 126181175, 136500277, 679840405, 735437783, 795581917
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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(-35,a(1)) and (A130013(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (649+36*sqrt(2))/647 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 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)=613, a(2)=647, a(3)=685, a(4)=2993, a(5)=3235, a(6)=3497.
G.f.: (1-x)*(613+1260*x+1945*x^2+1260*x^3+613*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 647*A001653(k) for k >= 1.
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EXAMPLE
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(-35, a(1)) = (-35, 613) is a solution: (-35)^2+(-35+647)^2 = 1225+374544 = 375769 = 613^2.
(A130013(1), a(2)) = (0, 647) is a solution: 0^2+(0+647)^2 = 418609 = 647^2.
(A130013(3), a(4)) = (1768, 2993) is a solution: 1768^2+(1768+647)^2 = 3125824+5832225 = 8958049 = 2993^2.
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PROGRAM
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(PARI) {forstep(n=-36, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609, &k), print1(k, ", ")))}
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CROSSREFS
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Cf. A130013, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).
Sequence in context: A022049 A107513 A151592 this_sequence A100364 A142435 A090869
Adjacent sequences: A159638 A159639 A159640 this_sequence A159642 A159643 A159644
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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 21 2009
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