A116697 a(n) = -a(n-1) - a(n-3) + a(n-4).

1, 1, -2, 2, -2, 5, -9, 13, -20, 34, -56, 89, -143, 233, -378, 610, -986, 1597, -2585, 4181, -6764, 10946, -17712, 28657, -46367, 75025, -121394, 196418, -317810, 514229, -832041, 1346269, -2178308, 3524578, -5702888

Offset: 0

Author

Creighton Dement, Feb 23 2006

Keywords

• easy
• nice
• sign

Links

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

Crossrefs

Cf. A000045, A001519, A006498, A056594, A115008, A116698, A116699, A128535.

Cf. A186679 (first differences).

Programs

Haskell

a116697 n = a116697_list !! n
a116697_list = [1,1,-2,2]
               ++ (zipWith (-) a116697_list
                               $ zipWith (+) (tail a116697_list)
                                             (drop 3 a116697_list))
a128535_list = 0 : (map negate $ map a116697 [0,2..])
a001519_list = 1 : map a116697 [1,3..]
a186679_list = zipWith (-) (tail a116697_list) a116697_list
a128533_list = map a186679 [0,2..]
a081714_list = 0 : (map negate $ map a186679 [1,3..])
a075193_list = 1 : -3 : (zipWith (+) a186679_list $ drop 2 a186679_list)
-- Reinhard Zumkeller, Feb 25 2011

Magma

A116697:= func< n | (-1)^Floor((n+1)/2)*(1+(-1)^n)/2 -(-1)^n*Fibonacci(n) >;
[A116697(n): n in [0..50]]; // G. C. Greubel, Jun 08 2025

Mathematica

LinearRecurrence[{-1,0,-1,1},{1,1,-2,2},40] (* Harvey P. Dale, Nov 04 2011 *)

SageMath

def A116697(n): return (-1)^(n//2)*((n+1)%2) - (-1)^n*fibonacci(n)
print([A116697(n) for n in range(51)]) # G. C. Greubel, Jun 08 2025

Formula

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

a(2*n+1) = A000045(2*n+1) = A001519(n).

a(2*n) = - A128535(n+1). - Reinhard Zumkeller, Feb 25 2011

a(n) = A056594(n) - (-1)^n*A000045(n). - Bruno Berselli, Feb 26 2011

E.g.f.: cos(x) + (2/sqrt(5))*exp(-x/2)*sinh(sqrt(5)*x/2). - G. C. Greubel, Jun 08 2025