login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A057682 a(n) = Sum((-1)^j*binomial(n,3*j+1),j=0..floor(n/3)). 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

Table of n, a(n) for n=0..38.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 13 14:58 EST 2017. Contains 295958 sequences.