A127968 a(n) = F(n+1) + (1-(-1)^n)/2, where F() = Fibonacci numbers A000045.

1, 2, 2, 4, 5, 9, 13, 22, 34, 56, 89, 145, 233, 378, 610, 988, 1597, 2585, 4181, 6766, 10946, 17712, 28657, 46369, 75025, 121394, 196418, 317812, 514229, 832041, 1346269, 2178310, 3524578, 5702888, 9227465, 14930353, 24157817, 39088170, 63245986, 102334156

Offset: 0

Author

Paul Barry, Feb 09 2007

Keywords

• easy
• nonn

Comments

Row sums of A127967.

The sequence beginning 1,1,2,2,4,... with g.f. x/(1-x-x^2) + 1/(1-x^2) has general term a(n) = F(n) + (1+(-1)^n)/2.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Programs

Magma

I:=[1,2,2,4]; [n le 4 select I[n] else Self(n-1) +2*Self(n-2) - Self(n-3) -Self(n-4): n in [1..30]]; // G. C. Greubel, May 04 2018

Mathematica

LinearRecurrence[{1,2,-1,-1},{1,2,2,4},40] (* Harvey P. Dale, Jun 19 2013 *)

PARI

Vec((1+x-2*x^2-x^3)/((1-x)*(1+x)*(1-x-x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

Formula

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

G.f.: (1 + x - 2*x^2 - x^3) / ((1 - x)*(1 + x)*(1 - x - x^2)).

From Colin Barker, Jul 12 2017: (Start)

a(n) = (5 - 5*(-1)^n + 2^(-n)*sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / 10.

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

(End)