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A159589 Positive numbers y such that y^2 is of the form x^2+(x+449)^2 with integer x. 4
421, 449, 481, 2045, 2245, 2465, 11849, 13021, 14309, 69049, 75881, 83389, 402445, 442265, 486025, 2345621, 2577709, 2832761, 13671281, 15023989, 16510541, 79682065, 87566225, 96230485, 464421109, 510373361, 560872369, 2706844589 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
(-29,a(1)) and (A130004(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (451+30*sqrt(2))/449 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 1.
LINKS
FORMULA
a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=421, a(2)=449, a(3)=481, a(4)=2045, a(5)=2245, a(6)=2465.
G.f.: (1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 449*A001653(k) for k >= 1.
EXAMPLE
(-29, a(1)) = (-29, 421) is a solution: (-29)^2+(-29+449)^2 = 841+176400 = 177241 = 421^2.
(A130004(1), a(2)) = (0, 449) is a solution: 0^2+(0+449)^2 = 201601 = 449^2.
(A130004(3), a(4)) = (1204, 2045) is a solution: 1204^2+(1204+449)^2 = 1449616+2732409 = 4182025 = 2045^2.
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {421, 449, 481, 2045, 2245, 2465}, 50] (* G. C. Greubel, May 08 2018 *)
PROG
(PARI) {forstep(n=-32, 50000000, [3, 1], if(issquare(2*n^2+898*n+201601, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4)/(1- 6*x^3+x^6)) \\ G. C. Greubel, May 08 2018
(Magma) I:=[421, 449, 481, 2045, 2245, 2465]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 08 2018
CROSSREFS
Cf. A130004, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).
Sequence in context: A028683 A198163 A014890 * A031784 A268859 A051648
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Apr 18 2009
STATUS
approved

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Last modified June 20 10:21 EDT 2024. Contains 373515 sequences. (Running on oeis4.)