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A057682 a(n) = Sum_{j=0..floor(n/3)} (-1)^j*binomial(n,3*j+1). 14
0, 1, 2, 3, 3, 0, -9, -27, -54, -81, -81, 0, 243, 729, 1458, 2187, 2187, 0, -6561, -19683, -39366, -59049, -59049, 0, 177147, 531441, 1062882, 1594323, 1594323, 0, -4782969, -14348907, -28697814, -43046721, -43046721, 0, 129140163, 387420489, 774840978 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1-M)^n = A057681(n)-a(n)*M+z(n)*M^2, where z(0)=z(1)=0 and, apparently, z(n+2)=A057083(n). - Stanislav Sykora, Jun 10 2012

From Tom Copeland, Nov 09 2014: (Start)

This array belongs to an interpolated family of arrays associated to the Catalan A000108 (t=1), and Riordan, or Motzkin sums A005043 (t=0), with the interp. (here t=-2) o.g.f. G(x,t) = x(1-x)/[1+(t-1)x(1-x)] and inverse o.g.f. Ginv(x,t) =  [1-sqrt(1-4x/(1+(1-t)x))]/2 (Cf. A005773 and A091867 and A030528 for more info on this family). (End)

{A057681, A057682, A*}, where A* is A057083 prefixed by two 0's, is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x)} of order 3. For the definitions of {k_i(x)} and the difference analog {K_i (n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Jul 31 2017

REFERENCES

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Mark W. Coffey, Reductions of particular hypergeometric functions 3F2 (a, a+1/3, a+2/3; p/3, q/3; +-1), arXiv preprint arXiv:1506.09160 [math.CA], 2015.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.

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

FORMULA

G.f.: (x - x^2) / (1 - 3*x + 3*x^2).

a(n) = 3*a(n-1) - 3*a(n-2), if n>1.

Starting at 1, the binomial transform of A000484. - Paul Barry, Jul 21 2003

It appears that abs(a(n)) = floor(abs(A000748(n))/3). - John W. Layman, Sep 05 2003

a(n) = ((3+i*sqrt(3))/2)^(n-2)+((3-i*sqrt(3))/2)^(n-2). - Benoit Cloitre, Oct 27 2003

a(n) = n*3F2(1/3-n/3,2/3-n/3,1-n/3 ; 2/3,4/3 ; 1) for n>=1. - John M. Campbell, Jun 01 2011

Let A(n) be the n X n matrix with -1's along the main diagonal, 1's everywhere above the main diagonal, and 1's along the subdiagonal. Then a(n) equals (-1)^(n-1) times the sum of the coefficients of the characteristic polynomial of A(n-1), for all n>1 (see Mathematica code below). - John M. Campbell, Mar 16 2012

Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1)=x(n)-z(n), y(n+1)=y(n)-x(n), z(n+1)=z(n)-y(n). Then a(n)=-y(n). But this recurrence falls into a repetitive cycle of length 6 and multiplicative factor -27, so that a(n) = -27*a(n-6) for any n>6. - Stanislav Sykora, Jun 10 2012

a(n) = A057083(n-1)-A057083(n-2). - R. J. Mathar, Oct 25 2012

G.f.: 3*x - 1/3 + 3*x/(G(0) - 1) where G(k)= 1 + 3*(2*k+3)*x/(2*k+1 - 3*x*(k+2)*(2*k+1)/(3*x*(k+2) + (k+1)/G(k+1)));(continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Nov 23 2012

G.f.: Q(0,u) -1, where u=x/(1-x), Q(k,u) = 1 - u^2 + (k+2)*u - u*(k+1 - u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013

From Vladimir Shevelev, Jul 31 2017: (Start)

For n>=1, a(n) = 2*3^((n-2)/2)*cos(Pi*(n-2)/6);

For n>=2, a(n) = K_1(n) + K_3(n-2);

For m,n>=2, a(n+m) = a(n)*K_1(m) + K_1(n)*a(m) - K_3(n-2)*K_3(m-2), where

K_1 = A057681, K_3 = A057083. (End)

EXAMPLE

x + 2*x^2 + 3*x^3 + 3*x^4 - 9*x^6 - 27*x^7 - 54*x^8 - 81*x^9 + ...

If M^3=1 then (1-M)^6 = A057681(6)-a(6)*M+A057083(4)*M^2 = -18+9*M+9*M^2. - Stanislav Sykora, Jun 10 2012

MAPLE

A057682:=n->add((-1)^j*binomial(n, 3*j+1), j=0..floor(n/3)):

seq(A057682(n), n=0..50); # Wesley Ivan Hurt, Nov 11 2014

MATHEMATICA

A[n_] := Array[KroneckerDelta[#1, #2 + 1] - KroneckerDelta[#1, #2] + Sum[KroneckerDelta[#1, #2 - q], {q, n}] &, {n, n}]; Join[{0, 1}, Table[(-1)^(n - 1)*Total[CoefficientList[CharacteristicPolynomial[A[(n - 1)], x], x]], {n, 2, 30}]] (* John M. Campbell, Mar 16 2012 *)

PROG

(PARI) {a(n) = sum( j=0, n\3, (-1)^j * binomial(n, 3*j + 1))} /* Michael Somos, May 26 2004 */

(PARI) {a(n) = if( n<2, n>0, n-=2; polsym(x^2 - 3*x + 3, n)[n + 1])} /* Michael Somos, May 26 2004 */

(MAGMA) I:=[0, 1, 2]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

CROSSREFS

Alternating row sums of triangle A030523.

Cf. A057681, A057083, A000108, A005043, A005773, A091867.

Sequence in context: A106242 A121474 A138003 * A124841 A085355 A103120

Adjacent sequences:  A057679 A057680 A057681 * A057683 A057684 A057685

KEYWORD

sign,easy

AUTHOR

N. J. A. Sloane, Oct 20 2000

STATUS

approved

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Last modified December 13 08:08 EST 2018. Contains 318082 sequences. (Running on oeis4.)