A116698 -id:A116698 - cp's OEIS frontend

A115008 a(n) = a(n-1) + a(n-3) + a(n-4).

1, 0, 2, 4, 5, 7, 13, 22, 34, 54, 89, 145, 233, 376, 610, 988, 1597, 2583, 4181, 6766, 10946, 17710, 28657, 46369, 75025, 121392, 196418, 317812, 514229, 832039, 1346269, 2178310, 3524578, 5702886, 9227465, 14930353, 24157817, 39088168

Offset: 0

Creighton Dement, Feb 23 2006

a(n+2) - a(n+1) - a(n) gives match to A000034, apart from signs.

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).

Cf. A000034, A000045, A001519, A001906, A006498, A116697, A116698, A116699.

A115008:= func< n | Fibonacci(n+1) - (n mod 2) + 2*0^((n+1) mod 4) >;
[A115008(n): n in [0..50]]; // G. C. Greubel, Aug 24 2025

Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2], {n,0,50}] (* G. C. Greubel, Aug 24 2025 *)

def A115008(n): return fibonacci(n+1) -i**(n-1)*(n%2)
print([A115008(n) for n in range(51)]) # G. C. Greubel, Aug 24 2025

a(2*n) = A000045(2*n+1) = A001519(n).
G.f.: (1-x+2*x^2+x^3)/((1+x^2)*(1-x-x^2)).
a(2*n+1) = (-1)^(n+1) + A001906(n+1) (compare with a similar property for A116697) - Creighton Dement, Mar 31 2006
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = A000045(n+1) - i^(n-1)*(n mod 2).
E.g.f.: exp(x/2)*(cosh(p*x) + (1/(2*p))*sinh(p*x)) - sin(x), where 2*p = sqrt(5). (End)

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

Original entry on oeis.org

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

Creighton Dement, Feb 23 2006

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).

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

Cf. A186679 (first differences).

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

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

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

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

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

cp's OEIS Frontend

A115008 a(n) = a(n-1) + a(n-3) + a(n-4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula