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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 04 2002
STATUS
approved