A106249 -id:A106249 - cp's OEIS frontend

A187180 Parse the infinite string 0101010101... into distinct phrases 0, 1, 01, 010, 10, ...; a(n) = length of n-th phrase.

1, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 13, 14, 15, 14, 15, 16, 17, 16, 17, 18, 19, 18, 19, 20, 21, 20, 21, 22, 23, 22, 23, 24, 25, 24, 25, 26, 27, 26, 27, 28, 29, 28, 29, 30, 31, 30, 31, 32, 33, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37, 38, 39, 38, 39, 40, 41, 40, 41, 42, 43, 42, 43, 44, 45, 44, 45, 46, 47, 46, 47, 48, 49, 48, 49, 50, 51, 50, 51, 52, 53, 52, 53, 54, 55, 54, 55, 56, 57, 56, 57, 58, 59, 58, 59, 60, 61

Offset: 1

N. J. A. Sloane, Mar 06 2011

			The sequence begins
   1   1
   2   3   2   3
   4   5   4   5
   6   7   6   7
   8   9   8   9
  10  11  10  11 ...

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

See A187180-A187188 for alphabets of size 2 through 10.

See also A109337, A187199, A187200, A106249, A083219, A018837.

1,1,seq(op(2*i*[1,1,1,1]+[0,1,0,1]), i=1..100); # Robert Israel, Oct 15 2015

Join[{1},LinearRecurrence[{1, 0, 0, 1, -1},{1, 2, 3, 2, 3},119]] (* Ray Chandler, Aug 26 2015 *)
CoefficientList[Series[(x^5 - 2 x^4 + x^3 + x^2 + 1)/((x - 1)^2 (x + 1) (x^2 + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Oct 16 2015 *)

a(n) = if(n==1, 1, (1 + (-1)^n + (1-I)*(-I)^n + (1+I)*I^n + 2*n) / 4); \\ Colin Barker, Oct 15 2015

Vec(x*(x^5-2*x^4+x^3+x^2+1) / ((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Oct 15 2015

Consider more generally the string 012...k012...k012...k012...k01... with an alphabet of size B, where k = B-1. The sequence begins with B 1's, and thereafter is quasi-periodic with period B^2, and increases by B in each period.
For the present example, where B=2, the sequence begins with two 1's and thereafter increases by 2 in each block of 4: (1,1) (2,3,2,3), (4,5,4,5), (6,7,6,7), ...
From Colin Barker, Oct 15 2015: (Start)
a(n) = (1+(-1)^n+(1-i)*(-i)^n+(1+i)*i^n+2*n)/4 for n>1, where i = sqrt(-1).
G.f.: x*(x^5-2*x^4+x^3+x^2+1) / ((x-1)^2*(x+1)*(x^2+1)). (End)
From Wesley Ivan Hurt, May 03 2021: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5).
a(n) = floor((n+1+(-1)^floor((n+1)/2))/2) for n > 1. (End)

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, 11, 6, 13, 8, 15, 8, 17, 10, 19, 10, 21, 12, 23, 12, 25, 14, 27, 14, 29, 16, 31, 16, 33, 18, 35, 18, 37, 20, 39, 20, 41, 22, 43, 22, 45, 24, 47, 24, 49, 26, 51, 26, 53, 28, 55, 28, 57, 30, 59, 30, 61, 32, 63, 32, 65, 34, 67, 34, 69, 36, 71, 36, 73, 38, 75

Offset: 0

Paul Curtz, Aug 14 2012

First differences: (1, 1, 1, -1, 3, -1, 3, -3, 5,...) = (1, A186422).
Second differences: (0, 0, -2, 4, -4, 4, -6, 8, ...)  = (-1)^(n+1) * A201629(n).
Interleave the terms with even indices of the companion A215495 and this one to get (A215495(0), A212831(0), A215495(2), A212831(2),...) = (1, 0, 1, 2, 3, 2, 3, 4, 5, 4,...) = A106249, up to the initial term = A083219 = A083220/2.

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

Cf. A214282, A129756.

[(1/4)*((1 +(-1)^n)*(1 - (-1)^Floor(n/2)) + (3 -(-1)^n)*n): n in [0..50]]; // G. C. Greubel, Apr 25 2018

a[n_] := (1/4)*((-(1 + (-1)^n))*(-1 + (-1)^Floor[n/2]) - (-3 + (-1)^n)*n ); Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Sep 18 2012 *)
LinearRecurrence[{0,1,0,1,0,-1},{0,1,2,3,2,5},80] (* Harvey P. Dale, May 29 2016 *)

A212831(n)=if(bittest(n,0), n, n\2+bittest(n,1)) \\ M. F. Hasler, Oct 21 2012

for(n=0,50, print1((1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n), ", ")) \\ G. C. Greubel, Apr 25 2018

a(n) + A215495(n) = A043547(n).
a(n) = -A214283(n)/A000108([n/2]).
a(n+1) = (A186421(n)=0,1,2,1,4,...) + 1.
a(2*n) = A052928(n+1).
a(n+2) - a(n) = 2, 2, 0, 2. (period 4).
a(n) = a(n-2) +a(n-4) -a(n-6); also holds for A215495(n).
G.f.: x*(1+2*x+2*x^2+x^4) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Aug 21 2012
a(n) = (1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n). - G. C. Greubel, Apr 25 2018

Corrected and edited by M. F. Hasler, Oct 21 2012

A110514 Expansion of (1 - x + x^2 + x^3)/(1 - x^2 - x^4 + x^6).

Original entry on oeis.org

1, -1, 2, 0, 3, -1, 4, 0, 5, -1, 6, 0, 7, -1, 8, 0, 9, -1, 10, 0, 11, -1, 12, 0, 13, -1, 14, 0, 15, -1, 16, 0, 17, -1, 18, 0, 19, -1, 20, 0, 21, -1, 22, 0, 23, -1, 24, 0, 25, -1, 26, 0, 27, -1, 28, 0, 29, -1, 30, 0, 31, -1, 32, 0, 33, -1, 34, 0, 35, -1, 36, 0, 37, -1, 38, 0, 39, -1, 40, 0, 41, -1, 42, 0, 43, -1, 44, 0, 45, -1, 46, 0, 47, -1, 48, 0

Offset: 0

Paul Barry, Jul 24 2005

Diagonal sums of A110515. Partial sums of A110516.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A106249.

Riffle[Range[50],{-1,0}] (* Harvey P. Dale, Dec 08 2011 *)

x='x+O('x^50); Vec((1-x+x^2+x^3)/((1-x^2)^2(1+x^2))) \\ G. C. Greubel, Aug 29 2017

G.f.: (1 - x + x^2 + x^3)/((1 - x^2)^2(1 + x^2)).
a(n) = a(n-2) + a(n-4) - a(n-6).
a(n) = (n/2 + 1)*(1 + (-1)^n)/2 - (1 - (-1)^n)*(1 + (-1)^((n-1)/2))/4.
a(n) = (sin(Pi*n/2)*((-1)^n - 1) + (n+3)*(-1)^n + (n+1))/4.
a(n) = Sum_{k=0..floor(n/2)} Jacobi(2^(n-2k), 2(n-2k)+1) [conjecture].

A110515 Sequence array for (1 - x + x^2 + x^3)/(1 - x^4).

Original entry on oeis.org

1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1

Offset: 0

Paul Barry, Jul 24 2005

Row sums are A106249. Diagonal sums are A110514.

			Rows begin
   1;
  -1,  1;
   1, -1,  1;
   1,  1, -1,  1;
   1,  1,  1, -1,  1;
  -1,  1,  1,  1, -1,  1;
   1, -1,  1,  1,  1,- 1,  1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Table[If[k <= n, -Sin[Pi*(n - k)/2] + Cos[Pi*(n - k)]/2 + 1/2, 0], {n,0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 29 2017 *)

for(n=0,20, for(k=0,n, print1(round(if(k<=n, -sin(Pi*(n-k)/2) + cos(Pi*(n-k))/2 + 1/2, 0)), ", "))) \\ G. C. Greubel, Aug 29 2017

Riordan array ((1 - x + x^2 + x^3)/(1 - x^4), 1).
Column k has g.f. x^k*(1 - x + x^2 + x^3)/(1 - x^4).
T(n, k) = if(k <= n, -sin(Pi*(n-k)/2) + cos(Pi*(n-k))/2 + 1/2, 0).
T(n, k) = if(k <= n, Jacobi(2^(n-k), 2(n-k)+1), 0) [conjecture].

A234013 Number of maximally biased free polyominoes with n squares.

Original entry on oeis.org

1, 1, 2, 1, 1, 11, 8, 3, 1, 79, 36, 8, 2, 540, 164, 31, 4, 3174, 749, 106, 11, 17443, 3312, 397, 27

Offset: 1

John Mason, Dec 27 2013

Define the bias of a polyomino to be the difference between the number of black squares and the number of white squares when chessboard coloring is applied to the polyomino. Maximally biased polyominoes of size n are those sharing the maximum value of bias among all polyominoes of n squares. For n = 4m + 1, for integer m, all maximally biased polyominoes may be built starting with a monomino and then successively adding "airplane" tetrominoes.

John Mason, Chessboard coloured polyominoes and bias
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).

Cf. A000105, A106249, A001933, A234012.

More terms from John Mason, Jan 03 2015

A106251 Expansion of (1-x+x^2+x^3+x^5)/(1-x-x^6+x^7).

Original entry on oeis.org

1, 0, 1, 2, 2, 3, 4, 3, 4, 5, 5, 6, 7, 6, 7, 8, 8, 9, 10, 9, 10, 11, 11, 12, 13, 12, 13, 14, 14, 15, 16, 15, 16, 17, 17, 18, 19, 18, 19, 20, 20, 21, 22, 21, 22, 23, 23, 24, 25, 24, 25, 26, 26, 27, 28, 27, 28, 29, 29, 30, 31, 30, 31, 32, 32, 33, 34, 33, 34, 35, 35, 36, 37, 36, 37, 38

Offset: 0

Paul Barry, Apr 27 2005

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A008611, A106249, A106250.

LinearRecurrence[{1,0,0,0,0,1,-1},{1,0,1,2,2,3,4},80] (* Harvey P. Dale, Oct 04 2021 *)

G.f.: (1+x^2+2x^3+2x^4+3x^5+2x^6+3x^7+2x^8+x^9+x^10)/(1-x^6)^2; a(n)=sum{k=0..n, -mu(k mod 6)}.

cp's OEIS Frontend

A187180 Parse the infinite string 0101010101... into distinct phrases 0, 1, 01, 010, 10, ...; a(n) = length of n-th phrase.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A212831 a(4*n) = 2*n, a(2*n+1) = 2*n+1, a(4*n+2) = 2*n+2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A110514 Expansion of (1 - x + x^2 + x^3)/(1 - x^2 - x^4 + x^6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A110515 Sequence array for (1 - x + x^2 + x^3)/(1 - x^4).

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

PARI

Formula

A234013 Number of maximally biased free polyominoes with n squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A106251 Expansion of (1-x+x^2+x^3+x^5)/(1-x-x^6+x^7).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.