A246715 n * Lucas(n) - (n - 1) * Lucas(n - 1).

1, 5, 6, 16, 27, 53, 95, 173, 308, 546, 959, 1675, 2909, 5029, 8658, 14852, 25395, 43297, 73627, 124909, 211456, 357270, 602551, 1014551, 1705657, 2863493, 4800990, 8039608, 13447563, 22469261, 37505879, 62546285, 104212364, 173489994, 288593903, 479706787, 796815125, 1322659237, 2194126122, 3637574444, 6027141411, 9980945785

Offset: 1

Author

Giuseppe Coppoletta, Sep 02 2014

Keywords

• nonn
• easy

Comments

By definition, the arithmetic mean of a(1), ... a(n) is equal to L(n) and a(n) - Lucas(n) = (n - 1) * Lucas(n - 2). See A136391 for the Fibonacci case.

Examples

			a(6) = 53 = 6*Lucas(6) - 5*Lucas(5) = 6 * 18 - 5 * 11 = 108 - 55.
a(4) = 16 = 4*Lucas(2) + Lucas(3) = 3*Lucas(2) + Lucas(4).
		

Crossrefs

Cf. A000032, A000045, A136391.

Programs

Maple

with(combinat): seq(n*(fibonacci(n-1)+fibonacci(n-3)) +fibonacci(n)+fibonacci(n-2),n=1..40).

Mathematica

Table[LucasL[n]n - LucasL[n - 1](n - 1), {n, 35}] (* Alonso del Arte, Sep 02 2014 *)

PARI

a(n) = n*(fibonacci(n-1)+fibonacci(n-3)) +fibonacci(n)+fibonacci(n-2); \\ Michel Marcus, Sep 02 2014

Formula

Recurrence: a(n + 1) = a(n) + a(n - 1) + 5*F(n - 2), n >= 2, where F = A000045. Proof: similar to A136391.

Also, a(n) = 2*a(n - 1) + a(n - 2) - 2*a(n - 3) - a(n - 4).

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