A117441 Periodic with repeating part {1,1,0,1,-1,0,-1,-1,0,-1,1,0}.

1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0, 1, 1, 0, 1, -1, 0

Offset

0,1

Comments

Diagonal sums of number triangle A117440.

Links

Table of n, a(n) for n=0..77.

Index entries for linear recurrences with constant coefficients, signature (0,1,0,-1).

Formula

G.f.: (1+x-x^2)/(1-x^2+x^4).

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(cos(Pi*(n-2*k)/2)+sin(Pi*(n-2*k)/2)).

a(1)=a(2)=1; a(n) = a(n-2) + (-1)^n*a(n-1). - José María Grau Ribas, Jan 08 2012

a(n) = A260190(n+2) = A260192(n-1). a(2*n + 1) = A010892(n). a(3*n) = A057077(n). a(3*n + 1) = A087960(n). a(3*n + 2) = 0. - Michael Somos, Jul 18 2015

Example

G.f. = 1 + x + x^3 - x^4 - x^6 - x^7 - x^9 + x^10 + x^12 + x^13 + x^15 + ...

Mathematica

a[1] := 1; a[2] := 1; a[n_] := a[n] = a[n - 2] + (-1)^(n) a[n - 1]; Array[a, 100] (* José María Grau Ribas, Jan 08 2012 *)
PadRight[{}, 84, {1, 1, 0, 1, -1, 0, -1, -1, 0, -1, 1, 0}] (* Harvey P. Dale, Mar 30 2012 *)
a[ n_] := KroneckerSymbol[ -6, 2 n + 5]; (* Michael Somos, Jul 18 2015 *)
LinearRecurrence[{0, 1, 0, -1}, {1, 1, 0, 1}, 78] (* Ray Chandler, Aug 25 2015 *)

Prog

(PARI) Vec((1+x-x^2)/(1-x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Jan 10 2012
(PARI) {a(n) = kronecker( -6, 2*n + 5)}; /* Michael Somos, Jul 18 2015 */

Crossrefs

Cf. A010892, A057077, A087960, A260190, A260192.

Sequence in context: A016118 A011646 A016350 * A049347 A010892 A091338

Adjacent sequences: A117438 A117439 A117440 * A117442 A117443 A117444

Keyword

easy,sign

Author

Paul Barry, Mar 16 2006

Extensions

More terms from Sean A. Irvine, Sep 26 2011

Status

approved