A112689 A modified Chebyshev transform of the Jacobsthal numbers.

0, 1, 1, 0, 1, 2, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 4, 3, 3, 4, 4, 4, 4, 4, 5, 5, 4, 5, 6, 5, 5, 6, 6, 6, 6, 6, 7, 7, 6, 7, 8, 7, 7, 8, 8, 8, 8, 8, 9, 9, 8, 9, 10, 9, 9, 10, 10, 10, 10, 10, 11, 11, 10, 11, 12, 11, 11, 12, 12, 12, 12, 12, 13, 13, 12, 13, 14, 13, 13, 14, 14, 14, 14, 14, 15, 15, 14

Offset: 0

Author

Paul Barry, Sep 15 2005

Keywords

• easy
• nonn

Examples

			G.f. = x + x^2 + x^4 + 2*x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,-1,2,-1,1,-1)

Crossrefs

Cf. A051275.

Programs

Magma

I:=[0, 1, 1, 0, 1, 2]; [n le 6 select I[n] else Self(n-1)-Self(n-2)+2*Self(n-3)-Self(n-4)+Self(n-5)-Self(n-6): n in [1..100]]; // Vincenzo Librandi, Aug 14 2013

Mathematica

CoefficientList[Series[x / ((1 + x^2) (1 + x + x^2) (1 - x)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 14 2013 *)
a[ n_] := If[n > 0, SeriesCoefficient[ x / (1 - x + x^2 - 2 x^3 + x^4 - x^5 + x^6), {x, 0, n}], SeriesCoefficient[ -x^5 / (1 - x + x^2 - 2 x^3 + x^4 - x^5 + x^6), {x, 0, -n}]] (* Michael Somos, Dec 17 2013 *)
LinearRecurrence[{1,-1,2,-1,1,-1},{0,1,1,0,1,2},100] (* Harvey P. Dale, Apr 18 2022 *)

PARI

a(n) = floor((n+4)/6+(1-(-1)^n)*(-1)^floor(n/2)/4); \\ Joerg Arndt, Aug 14 2013

PARI

{a(n) = if( n>0, polcoeff( x / (1 - x + x^2 - 2*x^3 + x^4 - x^5 + x^6) + x * O(x^n), n), polcoeff( -x^5 / (1 - x + x^2 - 2*x^3 + x^4 - x^5 + x^6) + x * O(x^-n), -n))} /* Michael Somos, Dec 11 2013 */

Formula

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

a(n) = sum{k=0..floor((n+2)/2), (-1)^(k+1)*C(n-k+2, k-1)*A001045(n-2k+2)}.

a(n) = floor((n+4)/6+(1-(-1)^n)*(-1)^floor(n/2)/4). - Tani Akinari, Aug 13 2013

G.f.: x / (1 - x + x^2 - 2*x^3 + x^4 - x^5 + x^6). - Michael Somos, Dec 11 2013

a(-4 - n) = -a(n). a(2*n) = floor( (n+2) / 3). a(2*n + 1) = A051275(n). a(6*n) = a(6*n - 2) = a(6*n - 4) = n. a(6*n + 1) - 1 = a(6*n - 3) = a(6*n - 7) = 2 * floor(n/2). - Michael Somos, Dec 11 2013

0 = a(n) - a(n-1) + a(n-2) - 2*a(n-3) + a(n-4) - a(n-5) + a(n-6) for all n in Z. - Michael Somos, Dec 11 2013

Euler transform of length 4 sequence [ 1, -1, 1, 1]. - Michael Somos, Dec 17 2013