A026625 a(n) = Sum_{k=0..floor(n/2)} A026615(n-k,k).

1, 1, 2, 4, 7, 13, 21, 36, 58, 96, 155, 253, 409, 664, 1074, 1740, 2815, 4557, 7373, 11932, 19306, 31240, 50547, 81789, 132337, 214128, 346466, 560596, 907063, 1467661, 2374725, 3842388, 6217114, 10059504, 16276619, 26336125, 42612745, 68948872, 111561618

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032, A000045, A026615, A026616, A026617, A026618, A026619.

Cf. A026620, A026621, A026622, A026623, A026624, A026956, A026957.

Cf. A026958, A026959, A026960.

Programs

Magma

[n eq 1 select 1 else 3*Fibonacci(n+1) - 2*Fibonacci(n) - (3+(-1)^n)/2: n in [0..40]]; // G. C. Greubel, Jun 16 2024

Mathematica

Join[{1,1},Table[Fibonacci[n-1]+LucasL[n]-(3+(-1)^n)/2,{n,2,40}]] (* or *) Join[{1,1},LinearRecurrence[{1,2,-1,-1},{2,4,7,13},40]] (* Harvey P. Dale, Sep 27 2011 *)

PARI

Vec((1 - x^2 + x^3 + x^4 + x^5) / ((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

SageMath

[3*fibonacci(n+1) -2*fibonacci(n) -(3+(-1)^n)//2 + int(n==1) for n in range(41)] # G. C. Greubel, Jun 16 2024

Formula

For n>1, a(n) = Fibonacci(n-1) + Lucas(n) - (3 + (-1)^n)/2. - Ralf Stephan, May 13 2004

From Colin Barker, Jul 12 2017: (Start)

G.f.: (1 - x^2 + x^3 + x^4 + x^5) / ((1 - x)*(1 + x)*(1 - x - x^2)).

a(n) = 2^(-1-n)*(-5*((-2)^n + 3*2^n) - (-15+sqrt(5))*(1+sqrt(5))^n + (1-sqrt(5))^n*(15+sqrt(5))) / 5 for n>1.

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>5.

(End)