A157646 Positive numbers y such that y^2 is of the form x^2 + (x+31)^2 with integer x.

25, 31, 41, 109, 155, 221, 629, 899, 1285, 3665, 5239, 7489, 21361, 30535, 43649, 124501, 177971, 254405, 725645, 1037291, 1482781, 4229369, 6045775, 8642281, 24650569, 35237359, 50370905, 143674045, 205378379, 293583149, 837393701

Offset

1,1

Comments

(-7,a(1)) and (A118674(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+31)^2 = y^2.

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

Lim_{n -> infinity} a(n)/a(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {0, 2}.

Lim_{n -> infinity} a(n)/a(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 1.

For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, 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+1, a(3)=6m^2-10m+4, a(4)=3p, a(5)=6m^2+10m+4, a(6)=40m^2-58m+21.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=2m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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)=25, a(2)=31, a(3)=41, a(4)=109, a(5)=155, a(6)=221.

G.f.: (1-x)*(25 + 56*x + 97*x^2 + 56*x^3 + 25*x^4)/(1 - 6*x^3 + x^6).

a(3*k-1) = 31*A001653(k) for k >= 1.

Example

(-7, a(1)) = (-7, 25) is a solution: (-7)^2+(-7+31)^2 = 49+576 = 625 = 25^2.
(A118674(1), a(2)) = (0, 31) is a solution: 0^2+(0+31)^2 = 961 = 31^2.
(A118674(3), a(4)) = (60, 109) is a solution: 60^2+(60+31)^2 = 3600+8281 = 11881 = 109^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {25, 31, 41, 109, 155, 221}, 40] (* Harvey P. Dale, Oct 12 2017 *)

Prog

(PARI) {forstep(n=-8, 840000000, [1, 3], if(issquare(2*n^2+62*n+961, &k), print1(k, ", ")))};
(Magma) I:=[25, 31, 41, 109, 155, 221]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..50]]; // G. C. Greubel, Mar 31 2018

Crossrefs

Cf. A118674, A001653, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31), A157648 (decimal expansion of (1539+850*sqrt(2))/31^2).

Sequence in context: A167324 A307283 A184042 * A198003 A039323 A043146

Adjacent sequences: A157643 A157644 A157645 * A157647 A157648 A157649

Keyword

nonn,easy

Author

Klaus Brockhaus, Mar 11 2009

Status

approved