A159750 Positive numbers y such that y^2 is of the form x^2+(x+47)^2 with integer x.

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

Offset

1,1

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) +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

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).

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

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.

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.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {37, 47, 65, 157, 235, 353}, 50] (* G. C. Greubel, May 22 2018 *)

Prog

(PARI) {forstep(n=-12, 100000000, [1, 3], if(issquare(2*n^2+94*n+2209, &k), print1(k, ", ")))};
(PARI) x='x+O('x^30); Vec((1-x)*(37+84*x+149*x^2+84*x^3+37*x^4)/(1 -6*x^3 +x^6)) \\ G. C. Greubel, May 22 2018
(Magma) I:=[37, 47, 65, 157, 235, 353]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 22 2018

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: A043954 A349057 A244356 * A108333 A231254 A218834

Adjacent sequences: A159747 A159748 A159749 * A159751 A159752 A159753

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 30 2009

Status

approved