cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A117575 Expansion of (1-x^3)/((1-x)*(1+2*x^2)).

Original entry on oeis.org

1, 1, -1, -2, 2, 4, -4, -8, 8, 16, -16, -32, 32, 64, -64, -128, 128, 256, -256, -512, 512, 1024, -1024, -2048, 2048, 4096, -4096, -8192, 8192, 16384, -16384, -32768, 32768, 65536, -65536, -131072, 131072, 262144, -262144, -524288, 524288
Offset: 0

Views

Author

Paul Barry, Mar 29 2006

Keywords

Comments

Row sums of A116949.
From Paul Curtz, Oct 24 2012: (Start)
b(n) = abs(a(n)) = A158780(n+1) = 1,1,1,2,2,4,4,8,8,8,... .
Consider the autosequence (that is a sequence whose inverse binomial transform is equal to the signed sequence) of the first kind of the example. Its numerator is A046978(n), its denominator is b(n). The numerator of the first column is A075553(n).
The denominator corresponding to the 0's is a choice.
The classical denominator is 1,1,1,2,1,4,4,8,1,16,16,32,1,... . (End)

Examples

			   0/1,  1/1    1/1,   1/2,   0/2,  -1/4,  -1/4,  -1/8, ...
   1/1,  0/1,  -1/2,  -1/2,  -1/4,   0/4,   1/8,   1/8, ...
  -1/1, -1/2,   0/2,   1/4,   1/4,   1/8,   0/8, -1/16, ...
   1/2,  1/2,   1/4,   0/4   -1/8,  -1/8, -1/16,  0/16, ...
   0/2, -1/4,  -1/4,  -1/8,   0/8,  1/16,  1/16,  1/32, ...
  -1/4,  0/4,   1/8,   1/8,  1/16,  0/16, -1/32, -1/32, ...
   1/4,  1/8,   0/8, -1/16, -1/16, -1/32,  0/32,  1/64, ...
  -1/8, -1/8, -1/16,  0/16,  1/32,  1/32,  1/64,  0/64. - _Paul Curtz_, Oct 24 2012

Links

Crossrefs

Cf. A016116, A046978, A075553, A116949, A122016, A191754/A192366.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

Programs

  • Magma
    [1] cat [(-1)^Floor(n/2)*2^Floor((n-1)/2): n in [1..50]]; // G. C. Greubel, Apr 19 2023
  • Mathematica
    CoefficientList[Series[(1-x^3)/((1-x)(1+2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{0,-2},{1,1,-1},45] (* Harvey P. Dale, Apr 09 2018 *)
  • PARI
    a(n)=if(n,(-1)^(n\2)<<((n-1)\2),1) \\ Charles R Greathouse IV, Jan 31 2012
  • SageMath
    def A117575(n): return 1 if (n==0) else (-1)^(n//2)*2^((n-1)//2)
[A117575(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

Formula

a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) for n >= 3.
a(n) = (cos(Pi*n/2) + sin(Pi*n/2)) * (2^((n-1)/2)*(1-(-1)^n)/2 + 2^((n-2)/2)*(1+(-1)^n)/2 + 0^n/2).
a(n+1) = Sum_{k=0..n} A122016(n,k)*(-1)^k. - Philippe Deléham, Jan 31 2012
E.g.f.: (1 + cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x))/2. - Stefano Spezia, Feb 05 2023
a(n) = (-1)^floor(n/2)*2^floor((n-1)/2), with a(0) = 1. - G. C. Greubel, Apr 19 2023

A116948 Riordan array ((1+2x^2)/(1-x^3),x).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1
Offset: 0

Views

Author

Paul Barry, Mar 29 2006

Keywords

Comments

Row sums are A117571. Diagonal sums are A117572. Inverse is A116949.

Examples

			Triangle begins
1,
0, 1,
2, 0, 1,
1, 2, 0, 1,
0, 1, 2, 0, 1,
2, 0, 1, 2, 0, 1,
1, 2, 0, 1, 2, 0, 1,
0, 1, 2, 0, 1, 2, 0, 1,
2, 0, 1, 2, 0, 1, 2, 0, 1

Formula

Number triangle T(n,k)=2*J(L((n-k+2)/3))[k<=n] where L(j/p) is the Legendre symbol of j and p and J(n)=A001045(n)

A117576 Expansion of (1-x^3)/((1-x^2)(1+2x^2)).

Original entry on oeis.org

1, 0, -1, -1, 3, 1, -5, -3, 11, 5, -21, -11, 43, 21, -85, -43, 171, 85, -341, -171, 683, 341, -1365, -683, 2731, 1365, -5461, -2731, 10923, 5461, -21845, -10923, 43691, 21845, -87381, -43691, 174763, 87381, -349525, -174763, 699051
Offset: 0

Views

Author

Paul Barry, Mar 29 2006

Keywords

Comments

A signed pair-reversal of the Jacobsthal numbers A001045. Diagonal sums of A116949.

Links

Crossrefs

Cf. A112447.

Formula

G.f.: (1+x+x^2)/(1+x+2x^2+2x^3); a(n)=-a(n-1)-2a(n-2)-2a(n-3); a(n)=2^(n/2)(2*cos(pi*n/2)/3+sqrt(2)*sin(pi*n/2)/6)+(-1)^n/3;
a(n) = floor(((-1)^(floor(n/2))*2^(2*floor(n/2)+1-floor((n+1)/2))+1)/3). - Tani Akinari, Nov 09 2012
Showing 1-3 of 3 results.

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