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

1,2

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).

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)

Mathematica

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

Prog

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

Crossrefs

Cf. A000032, A001622, A014217, A281362.

Sequence in context: A340658 A291443 A210145 * A164393 A164391 A164153

Adjacent sequences: A020953 A020954 A020955 * A020957 A020958 A020959

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms from Vladeta Jovovic, Apr 04 2002

Status

approved