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A159750
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Positive numbers y such that y^2 is of the form x^2+(x+47)^2 with integer x.
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3
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37, 47, 65, 157, 235, 353, 905, 1363, 2053, 5273, 7943, 11965, 30733, 46295, 69737, 179125, 269827, 406457, 1044017, 1572667, 2369005, 6084977, 9166175, 13807573, 35465845, 53424383, 80476433, 206710093, 311380123, 469051025, 1204794713
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| (-12, a(1)) and (A118675(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (51+14*sqrt(2))/47 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2=y^2 with p=m^2-2 a prime number in A028871, m>=2, the x values are given by the sequence defined by: a(n)=6*a(n-3)-a(n-6)+2p with a(1)=0, a(2)=2m+2, a(3)=3m^2-10m+8, a(4)=3p, a(5)=3m^2+10m+8, a(6)=20m^2-58m+42.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=m^2+2m+2, b(3)=5m^2-14m+10, b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 09 2009]
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).
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FORMULA
| a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=37, a(2)=47, a(3)=65, a(4)=157, a(5)=235, a(6)=353.
G.f.: (1-x)*(37+84*x+149*x^2+84*x^3+37*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 47*A001653(k) for k >= 1.
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EXAMPLE
| (-12, a(1)) = (-12, 37) is a solution: (-12)^2+(-12+47)^2 = 144+1225 = 1369 = 37^2.
(A118675(1), a(2)) = (0, 47) is a solution: 0^2+(0+47)^2 = 2209 = 47^2.
(A118675(3), a(4)) = (85, 157) is a solution: 85^2+(85+47)^2 = 7225+17424 = 24649 = 157^2.
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PROG
| (PARI) {forstep(n=-12, 100000000, [1, 3], if(issquare(2*n^2+94*n+2209, &k), print1(k, ", ")))}
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CROSSREFS
| Cf. A118675, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).
Sequence in context: A039351 A043174 A043954 * A108333 A039465 A083240
Adjacent sequences: A159747 A159748 A159749 * A159751 A159752 A159753
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KEYWORD
| nonn,easy
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AUTHOR
| Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 30 2009
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