A237133 Values of x in the solutions to x^2 - 3xy + y^2 + 19 = 0, where 0 < x < y.

4, 5, 7, 11, 17, 28, 44, 73, 115, 191, 301, 500, 788, 1309, 2063, 3427, 5401, 8972, 14140, 23489, 37019, 61495, 96917, 160996, 253732, 421493, 664279, 1103483, 1739105, 2888956, 4553036, 7563385, 11920003, 19801199, 31206973, 51840212, 81700916, 135719437

Offset: 1

Author

Colin Barker, Feb 04 2014

Keywords

• nonn
• easy

Comments

The corresponding values of y are given by a(n+2).

Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1216 = 0.

Examples

			11 is in the sequence because (x, y) = (11, 28) is a solution to x^2 - 3xy + y^2 + 19 = 0.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A001519, A005248, A055819, A237132, A218735.

Programs

Mathematica

LinearRecurrence[{0,3,0,-1},{4,5,7,11},40] (* Harvey P. Dale, Dec 15 2014 *)

PARI

Vec(-x*(x-1)*(4*x^2+9*x+4)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

Formula

a(n) = 3*a(n-2)-a(n-4).

G.f.: -x*(x-1)*(4*x^2+9*x+4) / ((x^2-x-1)*(x^2+x-1)).

a(n) = (1/2) * (F(n+4) + (-1)^n*F(n-5)), n>4, with F the Fibonacci numbers (A000045). - Ralf Stephan, Feb 05 2014