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Search: id:A159565
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| A159565 |
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Positive numbers y such that y^2 is of the form x^2+(x+241)^2 with integer x. |
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4
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| 221, 241, 265, 1061, 1205, 1369, 6145, 6989, 7949, 35809, 40729, 46325, 208709, 237385, 270001, 1216445, 1383581, 1573681, 7089961, 8064101, 9172085, 41323321, 47001025, 53458829, 240849965, 273942049, 311580889, 1403776469, 1596651269
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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(-21,a(1)) and (A129991(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 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)=221, a(2)=241, a(3)=265, a(4)=1061, a(5)=1205, a(6)=1369.
G.f.: (1-x)*(221+462*x+727*x^2+462*x^3+221*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 241*A001653(k) for k >= 1.
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EXAMPLE
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(-21, a(1)) = (-21, 221) is a solution: (-21)^2+(-21+241)^2 = 441+48400 = 48841 = 221^2.
(A129993(1), a(2)) = (0, 241) is a solution: 0^2+(0+241)^2 = 58081= 241^2.
(A129993(3), a(4)) = (620, 1061) is a solution: 620^2+(620+241)^2 = 384400+741321 = 1125721 = 1061^2.
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PROGRAM
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(PARI) {forstep(n=-24, 50000000, [3, 1], if(issquare(2*n^2+482*n+58081, &k), print1(k, ", ")))}
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CROSSREFS
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Cf. A129991, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).
Sequence in context: A140918 A049023 A091756 this_sequence A048931 A052478 A147284
Adjacent sequences: A159562 A159563 A159564 this_sequence A159566 A159567 A159568
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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 16 2009
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