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Search: id:A157213
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| A157213 |
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Positive numbers y such that y^2 is of the form x^2+(x+137)^2 with integer x. |
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5
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| 97, 137, 277, 305, 685, 1565, 1733, 3973, 9113, 10093, 23153, 53113, 58825, 134945, 309565, 342857, 786517, 1804277, 1998317, 4584157, 10516097, 11647045, 26718425, 61292305, 67883953, 155726393, 357237733, 395656673, 907639933
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OFFSET
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1,1
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COMMENT
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(-65, a(1)) and (A129544(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 1.
lim_{n -> infinity} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}.
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FORMULA
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a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=97, a(2)=137, a(3)=277, a(4)=305, a(5)=685, a(6)=1565.
G.f.: (1-x)*(97+234*x+511*x^2+234*x^3+97*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 137*A001653(k) for k >= 1.
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EXAMPLE
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(-65, a(1)) = (-65, 97) is a solution: (-65)^2+(-65+137)^2 = 4225+5184 = 9409 = 97^2.
(A129544(1), a(2)) = (0, 137) is a solution: 0^2+(0+137)^2 = 18769 = 137^2.
(A129544(3), a(4)) = (136, 305) is a solution: 136^2+(136+137)^2 = 18496+74529 = 93025 = 305^2.
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PROGRAM
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(PARI) {forstep(n=-68, 1000000000, [3, 1], if(issquare(n^2+(n+137)^2, &k), print1(k, ", ")))}
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CROSSREFS
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Cf. A129544, A001653, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))).
Sequence in context: A144325 A161367 A073076 this_sequence A000923 A142528 A139500
Adjacent sequences: A157210 A157211 A157212 this_sequence A157214 A157215 A157216
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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), Feb 25 2009
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