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A159896
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Positive numbers y such that y^2 is of the form x^2+(x+839)^2 with integer x.
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4
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785, 839, 901, 3809, 4195, 4621, 22069, 24331, 26825, 128605, 141791, 156329, 749561, 826415, 911149, 4368761, 4816699, 5310565, 25463005, 28073779, 30952241, 148409269, 163625975, 180402881, 864992609, 953682071, 1051465045
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OFFSET
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1,1
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COMMENTS
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(-56, a(1)) and (A130647(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (843+58*sqrt(2))/839 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 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>=5, the x values are given by the sequence defined by: a(n) = 6*a(n-3) -a(n-6) +2*p with a(1)=0, a(2) = 2*m+2, a(3) = 3*m^2 -10*m +8, a(4) = 3*p, a(5) = 3*m^2 +10*m +8, a(6) = 20*m^2 -58*m +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 +2*m +2, b(3)= 5*m^2 -14*m +10, b(4)= 5*p, b(5)= 5*m^2 +14*m +10, b(6)= 29*m^2 -82*m +58. - Mohamed Bouhamida, Sep 09 2009
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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)=785, a(2)=839, a(3)=901, a(4)=3809, a(5)=4195, a(6)=4621.
G.f.: (1-x)*(785+1624*x+2525*x^2+1624*x^3+785*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 839*A001653(k) for k >= 1.
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EXAMPLE
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(-56, a(1)) = (-56, 785) is a solution: (-56)^2+(-56+839)^2 = 3136+613089 = 616225 = 785^2.
(A130647(1), a(2)) = (0, 839) is a solution: 0^2+(0+839)^2 = 703921 = 839^2.
(A130647(3), a(4)) = (2241, 3809) is a solution: 2241^2+(2241+839)^2 = 5022081+9486400 = 14508481 = 3809^2.
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MATHEMATICA
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LinearRecurrence[{0, 0, 6, 0, 0, -1}, {785, 839, 901, 3809, 4195, 4621}, 30] (* Harvey P. Dale, Mar 03 2013 *)
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PROG
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(PARI) {forstep(n=-56, 10000000, [1, 3], if(issquare(2*n^2+1678*n+703921, &k), print1(k, ", ")))}
(Magma) I:=[785, 839, 901, 3809, 4195, 4621]; [n le 6 select I[n] else 6*Self(n-3) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 17 2018
(PARI) is(n, p=839)=for(m=sqrtint((max(n, 984)^2-p^2)\2)-p\2, n, m^2+(m+p)^2<n^2||return(m^2+(m+p)^2==n^2))
A159896(n)=(matrix(6, 6, i, j, if(i<6, i+1==j, j==4, 6, j==1, -1))^n*[785, 839, 901, 3809, 4195, 4621]~)[1] \\ M. F. Hasler, May 17 2018
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CROSSREFS
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Cf. A130647, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159897 (decimal expansion of (843+58*sqrt(2))/839), A159898 (decimal expansion of (1760979+1141390*sqrt(2))/839^2).
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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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