A136376 a(n) = n*F(n) + (n-1)*F(n-1).

1, 3, 8, 18, 37, 73, 139, 259, 474, 856, 1529, 2707, 4757, 8307, 14428, 24942, 42941, 73661, 125951, 214739, 365166, 619508, 1048753, 1771943, 2988457, 5031843, 8459504, 14201994, 23811349, 39873841, 66695539, 111440227, 186016962

Offset: 1

Author

Gary W. Adamson, Dec 28 2007

Keywords

• nonn
• easy

Comments

For n>2, mod 2 = (0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, ...), i.e., two evens followed by four odds (repeating).

Inverse binomial transform of A117202: (1, 4, 15, 52, ...). - Gary W. Adamson, Sep 03 2008

Examples

			a(5) = 37 = a(n)*F(n) + (n-1)*F(n-1) = 5*5 + 4*3 = 25 + 12.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

Cf. A000032, A000045, A045925, A128064, A117202, A238344.

Programs

Mathematica

Table[n*Fibonacci[n] + (n - 1)*Fibonacci[n - 1], {n, 1, 50}] (* Stefan Steinerberger, Dec 28 2007 *)

PARI

a(n)=n*fibonacci(n)+(n-1)*fibonacci(n-1) \\ Charles R Greathouse IV, Oct 07 2015

PARI

Vec(x*(1+x)*(1+x^2)/(x^2+x-1)^2 + O(x^100)) \\ Altug Alkan, Oct 28 2015

Formula

a(n) = n*F(n) + (n-1)*F(n-1). Equals the matrix product A128064 (unsigned) * A000045.

From R. J. Mathar, Jul 13 2009: (Start)

a(n) = A045925(n) + A045925(n-1).

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

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

a(n) = A238344(3n-2,n-1). - Alois P. Heinz, Apr 11 2014

From Vladimir Reshetnikov, Oct 28 2015: (Start)

a(n) = ((n+1)*F(n)+(n-1)*L(n))/2, where L(n) are Lucas numbers (A000032).

E.g.f.: (exp(phi*x)*(phi^3*x-1)-exp(-x/phi)*(phi^3+x)/phi)/(sqrt(5)*phi)+1, where phi=(1+sqrt(5))/2.

(End)

Extensions

More terms from Stefan Steinerberger, Dec 28 2007