A020956 a(n) = Sum_{k>=1} floor(tau^(n-k)) where tau is A001622.

1, 2, 4, 8, 14, 25, 42, 71, 117, 193, 315, 514, 835, 1356, 2198, 3562, 5768, 9339, 15116, 24465, 39591, 64067, 103669, 167748, 271429, 439190, 710632, 1149836, 1860482, 3010333, 4870830, 7881179, 12752025, 20633221, 33385263, 54018502, 87403783, 141422304

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Clark Kimberling, Problem 10520, Amer. Math. Mon. 103 (1996) p. 347.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Crossrefs

Cf. A000032, A001622, A014217, A281362.

Programs

Magma

I:=[1,2,4,8,14]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)-3*Self(n-3)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Nov 01 2016

Magma

[Lucas(n+1)-(2*n+5-(-1)^n)/4: n in [1..40]]; // G. C. Greubel, Apr 05 2024

Mathematica

LinearRecurrence[{2,1,-3,0,1}, {1,2,4,8,14}, 40] (* Vincenzo Librandi, Nov 01 2016 *)

PARI

Vec(x*(1-x^2+x^3)/((1-x-x^2)*(1+x)*(1-x)^2) + O(x^50)) \\ Michel Marcus, Nov 01 2016

Python

prpr = 0
prev = 1
for n in range(2,100):
    print(prev, end=", ")
    curr = prpr+prev + n//2
    prpr = prev
    prev = curr
# Alex Ratushnyak, Jul 30 2012

SageMath

[lucas_number2(n+1,1,-1) -(n+2+(n%2))//2 for n in range(1,41)] # G. C. Greubel, Apr 05 2024

Formula

G.f.: x*(1-x^2+x^3)/((1-x-x^2)*(1+x)*(1-x)^2). - Ralf Stephan, Apr 08 2004

a(n) = Lucas(n+1) - floor(n/2) - 1.

a(n) = Sum_{k=0..n-1} A014217(k).

a(n) = 2^(-2-n)*((-2)^n - 5*2^n + 2*(1-t)^(1+n) + 2*(1+t)^n + 2*t*(1+t)^n - 2^(1+n)*n) where t=sqrt(5). - Colin Barker, Feb 09 2017

From G. C. Greubel, Apr 05 2024: (Start)

a(n) = Lucas(n+1) - (1/4)*(2*n + 5 - (-1)^n).

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

Extensions

More terms from Vladeta Jovovic, Apr 04 2002