A076118 a(n) = sum_k {n/2<=k<=n} k * (-1)^(n-k) * C(k,n-k).

0, 1, 1, -1, -3, -2, 2, 5, 3, -3, -7, -4, 4, 9, 5, -5, -11, -6, 6, 13, 7, -7, -15, -8, 8, 17, 9, -9, -19, -10, 10, 21, 11, -11, -23, -12, 12, 25, 13, -13, -27, -14, 14, 29, 15, -15, -31, -16, 16, 33, 17, -17, -35, -18, 18, 37, 19, -19, -39, -20, 20, 41, 21, -21, -43, -22, 22, 45, 23, -23, -47, -24, 24, 49, 25, -25, -51, -26, 26

Offset

0,5

Comments

Piecewise linear depending on residue modulo 6. Might be described as an inverse Catalan transform of the nonnegative integers.

Number of compositions of n consisting of at most two parts, all congruent to {0,2} mod 3 (offset 1). - Vladeta Jovovic, Mar 10 2005

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Formula

a(n) = ( 2n*sin((n+1/2)*Pi/3) + sin(n*Pi/3)/sin(Pi/3) )/3. a(3n)=n*(-1)^n; a(3n+1)=(2n+1)*(-1)^n; a(3n+2)=(n+1)*(-1)^n.

a(n) = sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*(n-k)}. - Paul Barry, Nov 12 2004

Euler transform of length 6 sequence [ 1, -2, -2, 0, 0, 2]. - Michael Somos, Jul 14 2006

G.f.: x(1-x)/(1-x+x^2)^2 = x*(1-x^2)^2*(1-x^3)^2/((1-x)*(1-x^6)^2). a(-1-n)=a(n). - Michael Somos, Jul 14 2006

a(n+4) = 2*a(n+3)-3*a(n+2)+2*a(n+1)-a(n). - Robert Israel, Aug 07 2015

a(n) = A099254(n-1)-A099254(n-2). - R. J. Mathar, Apr 01 2018

Example

a(10) = -5*1 + 6*15 - 7*35 + 8*28 - 9*9 + 10*1 = -5 + 90 -245 + 224 - 81 + 10 = -7.

Maple

A076118:=n->add(k*(-1)^(n-k)*binomial(k, n-k), k=floor(n/2)..n); seq(A076118(n), n=0..50); # Wesley Ivan Hurt, May 08 2014
f:= gfun:-rectoproc({a(n+4) = 2*a(n+3)-3*a(n+2)+2*a(n+1)-a(n), a(0)=0, a(1)=1, a(2)=1, a(3)=-1}, a(n), remember):
map(f, [$0..100]); # Robert Israel, Aug 07 2015

Mathematica

Table[Sum[k*(-1)^(n - k)*Binomial[k, n - k], {k, Floor[n/2], n}], {n,
0, 50}] (* Wesley Ivan Hurt, May 08 2014 *)

Prog

(PARI) {a(n)=local(k=n%3); n=n\3; (-1)^n*((k>0)+n+(k==1)*n)} /* Michael Somos, Jul 14 2006 */
(PARI) {a(n)=if(n<0, n=-1-n); polcoeff(x*(1-x)/(1-x+x^2)^2+x*O(x^n), n)} /* Michael Somos, Jul 14 2006 */

Crossrefs

Cf. A038608(n)=a(3n)=-a(3n-1).

Cf. A078028, A099254 (partial sums).

See A151842 for a version without signs.

Sequence in context: A139420 A092895 A151842 * A309045 A210956 A282161

Adjacent sequences: A076115 A076116 A076117 * A076119 A076120 A076121

Keyword

sign,easy

Author

Henry Bottomley, Oct 31 2002

Status

approved