A157120 Positive numbers y such that y^2 is of the form x^2+(x+103)^2 with integer x.

73, 103, 205, 233, 515, 1157, 1325, 2987, 6737, 7717, 17407, 39265, 44977, 101455, 228853, 262145, 591323, 1333853, 1527893, 3446483, 7774265, 8905213, 20087575, 45311737, 51903385, 117078967, 264096157, 302515097, 682386227, 1539265205

Offset

1,1

Comments

(-48, a(1)) and (A157119(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

Links

Table of n, a(n) for n=1..30.

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) = 73, a(2) = 103, a(3) = 205, a(4) = 233, a(5) = 515, a(6) = 1157.

G.f.: x*(1-x)*(73+176*x+381*x^2+176*x^3+73*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 103*A001653(k) for k >= 1.

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

Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*sqrt(2))^2 for n mod 3 = 1.

Limit_{n -> oo} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}.

Example

(-48, a(1)) = (-48, 73) is a solution: (-48)^2+(-48+103)^2 = 2304+3025 = 5329 = 73^2.
(A157119(1), a(2)) = (0, 103) is a solution: 0^2+(0+103)^2 = 10609 = 103^2,
(A157119(3), a(4)) = (105, 233) is a solution: 105^2+(105+103)^2 = 11025+43264 = 54289 = 233^2.

Mathematica

Select[Table[Sqrt[x^2+(x+103)^2], {x, -50, 3*10^6}], IntegerQ] (* THe program generates the first 20 terms of the sequence. *) (* or *) LinearRecurrence[ {0, 0, 6, 0, 0, -1}, {73, 103, 205, 233, 515, 1157}, 50](* Harvey P. Dale, Aug 19 2020 *)

Prog

(PARI) {forstep(n=-48, 1100000000, [1, 3], if(issquare(2*n^2+206*n+10609, &k), print1(k, ", ")))}

Crossrefs

Cf. A157119, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

Sequence in context: A116210 A180522 A238679 * A118220 A144050 A087878

Adjacent sequences: A157117 A157118 A157119 * A157121 A157122 A157123

Keyword

nonn,easy

Author

Klaus Brockhaus, Feb 25 2009

Extensions

Typo corrected by Klaus Brockhaus, Mar 01 2009

Status

approved