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A159466
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Positive numbers y such that y^2 is of the form x^2 + (x+127)^2 with integer x.
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4
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113, 127, 145, 533, 635, 757, 3085, 3683, 4397, 17977, 21463, 25625, 104777, 125095, 149353, 610685, 729107, 870493, 3559333, 4249547, 5073605, 20745313, 24768175, 29571137, 120912545, 144359503, 172353217, 704729957, 841388843, 1004548165
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OFFSET
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1,1
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COMMENTS
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(-15, a(1)) and (A129992(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+127)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (129 + 16*sqrt(2))/127 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (34947 + 21922*sqrt(2))/127^2 for n mod 3 = 1.
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LINKS
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FORMULA
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a(n) = 6*a(n-3) - a(n-6)for n > 6; a(1)=113, a(2)=127, a(3)=145, a(4)=533, a(5)=635, a(6)=757.
G.f.: (1-x)*(113+240*x+385*x^2+240*x^3+113*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 127*A001653(k) for k >= 1.
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EXAMPLE
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(-15, a(1)) = (-15, 113) is a solution: (-15)^2 + (-15+127)^2 = 225 + 12544 = 12769 = 113^2.
(A129992(1), a(2)) = (0, 127) is a solution: 0^2 + (0+127)^2 = 16129 = 127^2.
(A129992(3), a(4)) = (308, 533) is a solution: 308^2 + (308+127)^2 = 94864 + 189225 = 284089 = 533^2.
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MATHEMATICA
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LinearRecurrence[{0, 0, 6, 0, 0, -1}, {113, 127, 145, 533, 635, 757}, 50] (* Harvey P. Dale, Feb 06 2015 *)
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PROG
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(PARI) {forstep(n=-16, 500000000, [1, 3], if(issquare(2*n^2+254*n+16129, &k), print1(k, ", ")))}
(Magma) I:=[113, 127, 145, 533, 635, 757]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jun 15 2018
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CROSSREFS
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Cf. A129992, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159467 (decimal expansion of (129+16*sqrt(2))/127), A159468 (decimal expansion of (34947+21922*sqrt(2))/127^2).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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