A341929 Bisection of the numerators of the convergents of cf (1,1,6,1,6,1,...,6,1).

1, 2, 15, 118, 929, 7314, 57583, 453350, 3569217, 28100386, 221233871, 1741770582, 13712930785, 107961675698, 849980474799, 6691882122694, 52685076506753, 414788729931330, 3265624762943887, 25710209373619766, 202416050226014241, 1593618192434494162, 12546529489249939055, 98778617721565018278

Offset: 0

Author

John O. Oladokun, Feb 23 2021

Keywords

• nonn
• easy

Comments

15*a(n)^2 - 11 is a square for all terms.

x = a(n) and y = a(n+1) satisfy the equation x^2 + y^2 - 8*x*y = -11.

x = a(n) and y = a(n+2) satisfy x^2 + y^2 - 62*x*y = -704.

Examples

			a(3) = 8*15 - 2 = 118.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (8,-1).

Crossrefs

After a(0), bisection of A237262.

Cf. A341927.

Programs

Mathematica

LinearRecurrence [{8, -1}, {1,2}, 15]

PARI

my(p=Mod('x,'x^2-8*'x+1)); a(n) = subst(lift(p^n),'x,2); \\ Kevin Ryde, Feb 27 2021

Formula

a(n) = 8*a(n-1) - a(n-2) for n >= 2.

a(n) = A237262(2*n) for n >= 1.

G.f.: (1 - 6*x)/(1 - 8*x + x^2). - Stefano Spezia, Mar 01 2021