A099324 -id:A099324 - cp's OEIS frontend

A049347 Period 3: repeat [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, 1, -1, 0

Offset: 0

Wolfdieter Lang

G.f. 1/cyclotomic(3, x) (the third cyclotomic polynomial).
Self-convolution yields (-1)^n*A099254(n). - R. J. Mathar, Apr 06 2008
Hankel transform of A099324. - Paul Barry, Aug 10 2009
A057083(n) = p(-1)  where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0..n. - Michael Somos, Apr 29 2012
a(n) appears, together with b(n) = A099837(n+3) in the formula 2*exp(2*Pi*n*I/3) = b(n) + a(n)*sqrt(3)*I, n >= 0, with I = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014
The binomial transform is 1, 0, -1, -1, 0, 1, 1, 0, -1, -1.. (see A010891). The inverse binom. transform is 1, -2, 3, -3, 0, 9, -27, 54, -81.. (see A057682). - R. J. Mathar, Feb 25 2023

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

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 175.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J.-P. Allouche and M. Mendès France, Stern-Brocot polynomials and power series, arXiv preprint arXiv:1202.0211 [math.NT], 2012.
Elena Barcucci, Antonio Bernini, Stefano Bilotta and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (-1,-1).
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to inverse of cyclotomic polynomials

Cf. A001006, A010892, A057078, A057083, A102283, A106510, A130716, A143818, A194770.

Alternating row sums of A049310 (Chebyshev-S). [Wolfdieter Lang, Nov 04 2011]

&cat[[1,-1,0]: n in [0..90]]; // Vincenzo Librandi, Apr 03 2014

A049347 := proc(n)
    op(modp(n,3)+1,[1,-1,0]) ;
end proc:
seq(A049347(n),n=0..100) ; # R. J. Mathar, Aug 06 2016

Flatten[Table[{1, -1, 0}, {27}]] (* Alonso del Arte, Feb 07 2013 *)
CoefficientList[Series[1/Cyclotomic[3, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
LinearRecurrence[{-1, -1}, {1, -1}, 90] (* Ray Chandler, Sep 15 2015 *)
Table[DirichletCharacter[3, 2, n + 1], {n, 0, 29}] (* Steven Foster Clark, May 29 2019 *)
Table[Mod[n + 2, 3] - 1, {n, 0, 20}] (* Michael Somos, Sep 24 2019 *)
Table[ChebyshevU[n, -1/2], {n, 0, 20}] (* Eric W. Weisstein, Jan 09 2024 *)
ChebyshevU[Range[0, 20], -1/2] (* Eric W. Weisstein, Jan 09 2024 *)

A049347(n) := block(
        [1,-1,0][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = n++; kronecker( -3, n)} /* Michael Somos, Oct 03 2006 */

{a(n) = [1, -1, 0][n%3 + 1]} /* Michael Somos, Oct 15 2008 */

a(n)=(n+2)%3-1 /* Jaume Oliver Lafont, Mar 24 2009 */

def A049347():
    x, y = 1, -1
    while True:
        yield x
        x, y = y, -x - y
a = A049347(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013

G.f.: 1/(1+x+x^2).
a(n) = +1 if n mod 3 = 0, a(n) = -1 if n mod 3 = 1, else 0.
a(n) = S(n, -1) = U(n, -1/2) (Chebyshev's U(n, x) polynomials.)
a(n) = 2*sqrt(3)*cos(2*Pi*n/3 + Pi/6)/3. - Paul Barry, Mar 15 2004
a(n) = Sum_{k >= 0} (-1)^(n-k)*C(n-k, k).
Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 - v + 2*u*v. - Michael Somos, Oct 03 2006
Euler transform of length 3 sequence [-1, 0, 1]. - Michael Somos, Oct 03 2006
a(n) = b(n+1) where b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = 1 if p == 1 (mod 3), b(p^e) = (-1)^e if p == 2 (mod 3). - Michael Somos, Oct 03 2006
From Michael Somos, Oct 03 2006: (Start)
G.f.: (1 - x) /(1 - x^3).
a(n) = -a(1-n) = -a(n-1) - a(n-2) = a(n-3). (End)
From Michael Somos, Apr 29 2012: (Start)
G.f.: 1 / (1 + x / ( 1 - x / (1 + x))).
a(n) = (-1)^n * A010892(n).
a(n) * n! = A194770(n+1).
Revert transform of A001006. Convolution inverse of A130716. MOBIUS transform of A002324. EULER transform is A111317. BIN1 transform of itself. STIRLING transform is A143818(n+2). (End)
a(-n) = A057078(n). a(n) = A106510(n+1) unless n=0. - Michael Somos, Oct 15 2008
G.f. A(x) = 1/(1+x+x^2) = Q(0); Q(k) =  1- x/(1 - x^2/(x^2 - 1 + x/(x - 1 + x^2/(x^2 - 1/Q(k+1))))); (continued fraction 3 kind, 5-step ). - Sergei N. Gladkovskii, Jun 19 2012
a(n) = -1 + floor(67/333*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = -1 + floor(19/26*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 03 2013
a(n) = ceiling((n-1)/3) - ceiling(n/3) + floor(n/3) - floor((n-1)/3). - Wesley Ivan Hurt, Dec 06 2013
a(n) = n + 1 - 3*floor((n+2)/3). - Mircea Merca, Feb 04 2014
a(n) = A102283(n+1) for all n in Z. - Michael Somos, Sep 24 2019
E.g.f.: exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Oct 26 2022

Edited by Charles R Greathouse IV, Mar 23 2010

A091491 Triangle, read by rows, where the n-th diagonal is generated from the n-th row by the sum of the products of the n-th row terms with binomial coefficients.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 8, 4, 1, 1, 23, 22, 13, 5, 1, 1, 65, 64, 41, 19, 6, 1, 1, 197, 196, 131, 67, 26, 7, 1, 1, 626, 625, 428, 232, 101, 34, 8, 1, 1, 2056, 2055, 1429, 804, 376, 144, 43, 9, 1, 1, 6918, 6917, 4861, 2806, 1377, 573, 197, 53, 10, 1

Offset: 0

Paul D. Hanna, Jan 14 2004

Row sums are A014137 (partial sums of Catalan numbers A000108). Columns equal the partial sums of the columns of the Catalan convolution triangle A033184. Columns include A014137, A014138, A001453.
Apart from the first column, any term is the partial sum of terms of the row above, when summing from the right. - Ralf Stephan, Apr 27 2004
Matrix inverse equals triangle A104402.
Riordan array (1/(1-x), x*c(x)) where c(x) is the g.f. of A000108. - Philippe Deléham, Nov 04 2009

			T(7,3) = T(4,0)*C(2,2) + T(4,1)*C(3,2) + T(4,2)*C(5,2) + T(4,3)*C(6,2) = (1)*1 + (4)*3 + (3)*6 + (1)*10 = 41.
Rows begin:
  1;
  1,     1;
  1,     2,     1;
  1,     4,     3,     1;
  1,     9,     8,     4,     1;
  1,    23,    22,    13,     5,     1;
  1,    65,    64,    41,    19,     6,    1;
  1,   197,   196,   131,    67,    26,    7,    1;
  1,   626,   625,   428,   232,   101,   34,    8,    1;
  1,  2056,  2055,  1429,   804,   376,  144,   43,    9,   1;
  1,  6918,  6917,  4861,  2806,  1377,  573,  197,   53,  10,  1;
  1, 23714, 23713, 16795,  9878,  5017, 2211,  834,  261,  64, 11,  1;
  1, 82500, 82499, 58785, 35072, 18277, 8399, 3382, 1171, 337, 76, 12, 1;
  ...
As to the production matrix M, top row of M^3 = [1, 4, 3, 1, 0, 0, 0, ...].

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Cf. A000108, A001453, A014137, A014138, A033184, A104402.

Cf. A096465 (reversed).

a091491 n k = a091491_tabl !! n !! k
a091491_row n = a091491_tabl !! n
a091491_tabl = iterate (\row -> 1 : scanr1 (+) row) [1]
-- Reinhard Zumkeller, Jul 12 2012

A091491:= func< n,k | k eq 0 select 1 else k*(&+[Binomial(2*(n-j)-k-1, n-j-1)/(n-j): j in [0..n-k]]) >;
[A091491(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2021

nmax = 11; t[n_, k_] := k*(2n-k-1)!*HypergeometricPFQ[{1, k-n, -n}, {k/2-n+1/2, k/2-n+1}, 1/4]/(n!*(n-k)!); t[, 0] = 1; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]] (* _Jean-François Alcover, Nov 14 2011, after Vladimir Kruchinin *)

T(n,k)=if(k>n || n<0 || k<0,0,if(k==0 || k==n,1, sum(j=0,n-k,T(n-k,j)*binomial(k+j-1,k-1)););)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff(2/(2-Y*(1-sqrt(1-4*X)))/(1-X),n,x),k,y)
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

PARI
```
T(n,k)=if(n
				
```

def A091491(n,k): return 1 if (k==0) else k*sum(binomial(2*(n-j)-k-1, n-j-1)/(n-j) for j in (0..n-k))
flatten([[A091491(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021

T(n, k) = Sum_{j=0..n-k} T(n-k, j)*C(k+j-1, k-1).
G.f.: 2/(2-y*(1-sqrt(1-4*x)))/(1-x).
T(n, k) = T(n-1, k-1) + T(n, k+1) for n>0, with T(n, 0)=1.
Recurrence: for k>0, T(n, k) = Sum_{j=k..n} T(n-1, j). - Ralf Stephan, Apr 27 2004
T(n+2,2)= |A099324(n+2)|. - Philippe Deléham, Nov 25 2009
T(n,k) = k * Sum_{i=0..n-k} binomial(2*(n-i)-k-1, n-i-1)/(n-i) for k>0; T(n,0)=1. - Vladimir Kruchinin, Feb 07 2011
From Gary W. Adamson, Jul 26 2011: (Start)
The n-th row of the triangle is the top row of M^n, where M is the following infinite square production matrix in which a column of (1,0,0,0,...) is prepended to an infinite lower triangular matrix of all 1's and the rest zeros:
  1, 1, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, ...
  0, 1, 1, 1, 0, 0, ...
  0, 1, 1, 1, 1, 0, ...
  0, 1, 1, 1, 1, 1, ...
(End)
Sum_{k=0..n} T(n,k) = Sum_{j=0..n} A000108(j) = A014137(n). - G. C. Greubel, Apr 30 2021

cp's OEIS Frontend

A049347 Period 3: repeat [1, -1, 0].

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

PARI

Sage

Formula

Extensions

A091491 Triangle, read by rows, where the n-th diagonal is generated from the n-th row by the sum of the products of the n-th row terms with binomial coefficients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

PARI

Sage

Formula