A237132 Values of x in the solutions to x^2 - 3xy + y^2 + 11 = 0, where 0 < x < y.

3, 4, 5, 9, 12, 23, 31, 60, 81, 157, 212, 411, 555, 1076, 1453, 2817, 3804, 7375, 9959, 19308, 26073, 50549, 68260, 132339, 178707, 346468, 467861, 907065, 1224876, 2374727, 3206767, 6217116, 8395425, 16276621, 21979508, 42612747, 57543099, 111561620

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 + 704 = 0.

Examples

			9 is in the sequence because (x, y) = (9, 23) is a solution to x^2 - 3xy + y^2 + 11 = 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, A237133, A218735.

Programs

PARI

Vec(-x*(x-1)*(3*x^2+7*x+3)/((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)*(3*x^2+7*x+3) / ((x^2-x-1)*(x^2+x-1)).

a(n) = F(n+2) + (-1)^n*F(n-3), n>1, with F the Fibonacci numbers (A000045). - Ralf Stephan, Feb 05 2014

Let h(n) = hypergeom([(1 - n)/2, n mod 2 - n/2], [1 - n], -4) then a(n) = h(n-1) + h(n) for n > 3. - Peter Luschny, Sep 03 2019