A079757 -id:A079757 - cp's OEIS frontend

A100219 Expansion of (1-2x)/((1-x)(1-x+x^2)).

1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0

Offset: 0

Paul Barry, Nov 08 2004

Row sums of number triangle A100218.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Cf. A079757, A100218.

&cat[[1,0,-2,-3,-2,0]: n in [0..20]]; // G. C. Greubel, Mar 28 2024

PadRight[{}, 120, {1,0,-2,-3,-2,0}] (* or *) LinearRecurrence[{2,-2,1}, {1,0,-2}, 50] (* G. C. Greubel, Mar 13 2017; Mar 28 2024 *)
Table[Cos[Pi*n/3 + Pi/3] + Sqrt[3]*Sin[Pi*n/3 + Pi/3] - 1, {n, 0, 71}] (* Indranil Ghosh, Mar 13 2017 *)

my(x='x+O('x^50)); Vec((1-2*x)/((1-x)*(1-x+x^2))) \\ G. C. Greubel, Mar 13 2017

def A100219(n): return [1,0,-2,-3,-2,0][n%6]
[A100219(n) for n in range(121)] # G. C. Greubel, Mar 28 2024

a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3).
a(n) = cos(Pi*n/3 + Pi/3) + sqrt(3)*sin(Pi*n/3 + Pi/3) - 1.
a(n) is the n-th order Taylor polynomial (centered at 0) of 1/c(x)^n evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. - Peter Bala, Apr 20 2024

A098599 Riordan array ((1+2*x)/(1+x), (1+x)).

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 1, 0, 3, 1, -1, 0, 2, 4, 1, 1, 0, 0, 5, 5, 1, -1, 0, 0, 2, 9, 6, 1, 1, 0, 0, 0, 7, 14, 7, 1, -1, 0, 0, 0, 2, 16, 20, 8, 1, 1, 0, 0, 0, 0, 9, 30, 27, 9, 1, -1, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 1, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, -1, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1, 1, 0, 0, 0, 0, 0, 0, 13, 91, 182, 156, 65, 13, 1

Offset: 0

Paul Barry, Sep 17 2004

			Triangle begins as:
   1;
   1, 1;
  -1, 2, 1;
   1, 0, 3, 1;
  -1, 0, 2, 4, 1;
   1, 0, 0, 5, 5,  1;
  -1, 0, 0, 2, 9,  6,  1;
   1, 0, 0, 0, 7, 14,  7,  1;
  -1, 0, 0, 0, 2, 16, 20,  8, 1;
   1, 0, 0, 0, 0,  9, 30, 27, 9, 1;

G. C. Greubel, Table of n, a(n) for n = 0..1325

Cf. A000007, A005408, A029635, A040000, A079757, A098601.
Row sums are A098600.
Diagonal sums are A098601.
Apart from signs, same as A100218.
Very similar to triangle A111125.

A098599:= func< n,k | n eq 0 select 1 else Binomial(k, n-k) + Binomial(k-1, n-k-1) >;
[A098599(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 27 2024

T[n_, k_]:= If[n==0, 1, Binomial[k,n-k] +Binomial[k-1,n-k-1]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 27 2024 *)

def A098599(n,k): return 1 if n==0 else binomial(k, n-k) + binomial(k-1, n-k-1)
flatten([[A098599(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 27 2024

Triangle: T(n, k) = binomial(k, n-k) + binomial(k-1, n-k-1), with T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = A098600(n) (row sums).
T(n,k) = T(n-1,k-1) - T(n-1,k) + 2*T(n-2,k-1) + T(n-3,k-1), T(0,0)=1, T(1,0)=1, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 09 2014
From G. C. Greubel, Mar 27 2024: (Start)
T(2*n, n) = A040000(n).
T(2*n+1, n) = A000007(n).
T(2*n-1, n) = A005408(n-1), n >= 1.
Sum_{k=0..n} (-1)^k*T(n, k) = A079757(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A098601(n). (End)

A131290 1 followed by period 6: repeat [3, 2, 0, -1, 0, 2].

Original entry on oeis.org

1, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1, 0, 2, 3, 2, 0, -1

Offset: 0

Paul Curtz, Sep 29 2007

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

Cf. A079757, A100219, A130869.

[1] cat &cat [[3, 2, 0, -1, 0, 2]^^30]; // Wesley Ivan Hurt, Jun 20 2016

A131290 := proc(n) if n = 0 then 1; else op(((n-1)mod 6)+1,[3,2,0,-1,0,2]) ; fi ; end: seq(A131290(n),n=0..100) ; # R. J. Mathar, Feb 27 2008

PadRight[{1},110,{2,3,2,0,-1,0}] (* or *) Join[{1},LinearRecurrence[ {2,-2,1},{3,2,0},110]] (* Harvey P. Dale, Jun 22 2012 *)

G.f.: (1+x-2*x^2+x^3)/((1-x)*(1-x+x^2)). - R. J. Mathar, Feb 27 2008
If n mod 6 = 4 then a(n) = (Fibonacci(n-3)*Fibonacci(n+1)) mod 4 -2, else a(n) = (Fibonacci(n-3)*Fibonacci(n+1)) mod 4, n>0. - Gary Detlefs, Dec 12 2010
From Wesley Ivan Hurt, Jun 20 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3) for n>4.
a(0) = 1, a(n) = 1 + cos(n*Pi/3) + sqrt(3)*sin(n*Pi/3) for n>0. (End)

More terms from R. J. Mathar, Feb 27 2008

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A100219 Expansion of (1-2x)/((1-x)(1-x+x^2)).

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A098599 Riordan array ((1+2*x)/(1+x), (1+x)).

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A131290 1 followed by period 6: repeat [3, 2, 0, -1, 0, 2].

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cp's OEIS Frontend

A100219 Expansion of (1-2*x)/((1-x)*(1-x+x^2)).

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Magma

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A098599 Riordan array ((1+2*x)/(1+x), (1+x)).

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Author

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Programs

Magma

Mathematica

SageMath

Formula

A131290 1 followed by period 6: repeat [3, 2, 0, -1, 0, 2].

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Magma

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Mathematica

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A100219 Expansion of (1-2x)/((1-x)(1-x+x^2)).