A026636 a(n) = Sum_{k=0..floor(n/2)} A026626(n-k, k).

1, 1, 2, 4, 6, 11, 17, 30, 47, 78, 125, 205, 330, 536, 866, 1404, 2270, 3675, 5945, 9622, 15567, 25190, 40757, 65949, 106706, 172656, 279362, 452020, 731382, 1183403, 1914785, 3098190, 5012975, 8111166, 13124141, 21235309

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,-1,-1).

Crossrefs

Cf. A026626, A026627, A026628, A026629, A026630, A026631, A026632.

Cf. A026633, A026634, A026635, A026961, A026962, A026963, A026964.

Cf. A026965.

Programs

Magma

I:=[2,4,6,11,17,30]; [1,1] cat [n le 6 select I[n] else Self(n-1) +Self(n-2) +Self(n-4) -Self(n-5) -Self(n-6): n in [1..50]]; // G. C. Greubel, Jun 21 2024

Mathematica

LinearRecurrence[{1,1,0,1,-1,-1}, {1,1,2,4,6,11,17,30}, 41] (* G. C. Greubel, Jun 21 2024 *)

SageMath

@CachedFunction
def a(n): # a = A026636
    if n<8: return (1,1,2,4,6,11,17,30)[n]
    else: return a(n-1) +a(n-2) +a(n-4) -a(n-5) -a(n-6)
[a(n) for n in range(41)] # G. C. Greubel, Jun 21 2024

Formula

G.f.: (1 + x^3 - x^4 + x^5 + x^7)/((1-x^4)*(1-x-x^2)).

From G. C. Greubel, Jun 21 2024: (Start)

a(n) = (1/20)*(2*LucasL(n-1) + 70*Fibonacci(n-1) - 15*(1+(-1)^n) - 4*cos((n-1)*Pi/2) - 2*sin((n-1)*Pi/2)) - [n=0] + [n=1].

E.g.f.: (1/10)*(cos(x) - 2*sin(x) - 15*cosh(x) - 10*(1 - x) + 2*exp(x/2)*(17*cosh(sqrt(5)*x/2) - 3*sqrt(5)*sinh(sqrt(5)*x/2))). (End)