A159809 Positive numbers y such that y^2 is of the form x^2+(x+223)^2 with integer x.

197, 223, 257, 925, 1115, 1345, 5353, 6467, 7813, 31193, 37687, 45533, 181805, 219655, 265385, 1059637, 1280243, 1546777, 6176017, 7461803, 9015277, 35996465, 43490575, 52544885, 209802773, 253481647, 306254033, 1222820173, 1477399307

Offset

1,1

Comments

(-28, a(1)) and (A130609(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.

Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

Lim_{n -> infinity} a(n)/a(n-1) = (227+30*sqrt(2))/223 for n mod 3 = {0, 2}.

Lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 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) = 5m^2 + 14*m + 10, b(6) = 29*m^2 - 82*m + 58. - Mohamed Bouhamida, Sep 09 2009

Links

G. C. Greubel, Table of n, a(n) for n = 1..3900

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=197, a(2)=223, a(3)=257, a(4)=925, a(5)=1115, a(6)=1345.

G.f.: (1-x)*(197+420*x+677*x^2+420*x^3+197*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 223*A001653(k) for k >= 1.

Example

(-28, a(1)) = (-28, 197) is a solution: (-28)^2 + (-28+223)^2 = 784 + 38025 = 38809 = 197^2.
(A130609(1), a(2)) = (0, 223) is a solution: 0^2 + (0+223)^2 = 49729 = 223^2.
(A130609(3), a(4)) = (533, 925) is a solution: 533^2 + (533+223)^2 = 284089 + 571536 = 855625 = 925^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {197, 223, 257, 925, 1115, 1345}, 50] (* G. C. Greubel, May 21 2018 *)

Prog

(PARI) {forstep(n=-28, 10000000, [1, 3], if(issquare(2*n^2+446*n+49729, &k), print1(k, ", ")))};
(Magma) I:=[197, 223, 257, 925, 1115, 1345]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 21 2018

Crossrefs

Cf. A130609, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A130610 (decimal expansion of (227+30*sqrt(2))/223), A130611 (decimal expansion of (105507+65798*sqrt(2))/223^2).

Sequence in context: A182572 A152625 A082246 * A345551 A051371 A127339

Adjacent sequences: A159806 A159807 A159808 * A159810 A159811 A159812

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 30 2009

Status

approved