A116697 a(n) = cos(n*Pi/2)-F(n)*(-1)^n, where F(n) = Fibonacci(n).

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

0,3

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).

Formula

a(n) = -a(n-1) - a(n-3) + a(n-4) for n > 3, with a(0) = a(1) = 1, a(2) = -2, a(3) = 2. (original name).

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

Mathematica

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

Prog

(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
(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

Crossrefs

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

Cf. A186679 (first differences).

Sequence in context: A339613 A008295 A216694 * A014244 A216634 A208054

Adjacent sequences: A116694 A116695 A116696 * A116698 A116699 A116700

Keyword

easy,nice,sign,changed

Author

Creighton Dement, Feb 23 2006

Extensions

New name from Wesley Ivan Hurt, Jun 12 2026

Status

approved