A191754 -id:A191754 - cp's OEIS frontend

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

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

Paul Barry, Mar 29 2006

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)

			   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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-2).

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

[1] cat [(-1)^Floor(n/2)*2^Floor((n-1)/2): n in [1..50]]; // G. C. Greubel, Apr 19 2023

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 *)

a(n)=if(n,(-1)^(n\2)<<((n-1)\2),1) \\ Charles R Greathouse IV, Jan 31 2012

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

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

A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2m-1,2m+1), where m = floor((n-2)/4).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 17, 28, 45, 72, 116, 188, 305, 494, 799, 1292, 2090, 3382, 5473, 8856, 14329, 23184, 37512, 60696, 98209, 158906, 257115, 416020, 673134, 1089154, 1762289, 2851444, 4613733, 7465176, 12078908, 19544084, 31622993, 51167078, 82790071

Offset: 2

Clark Kimberling

Essentially both the first difference sequence and partial sum of A005252, so its own shifted second difference and indeed virtually the same as A005252, so close to being its own shifted first difference.
From Paul Curtz, Jun 22 2011: (Start)
b(n) = 0,0,0,1,2,3,4,6, and differences are
   0,  0,  0,  1,  2,  3,  4,  6,
   0,  0,  1,  1,  1,  1,  2,  4,
   0,  1,  0,  0,  0,  1,  2,  3,
   1, -1,  0,  0,  1,  1,  1,  1,
  -2,  1,  0,  1,  0,  0,  0,  1,
   3, -1,  1  -1,  0,  0,  1,  1,
  -4,  2, -2,  1,  0,  1,  0,  0,
   6, -4,  3, -1,  1, -1,  0,  0;
b(n) is an autosequence (sequence identical to its inverse binomial transform signed) of first kind, i.e., its main diagonal is A000004.
Examples: A000045, A001045, A113405, A191754 (array). (End)
a(n) is the number of vertices of the Fibonacci cube Gamma(n-1) having an odd number of ones. The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1. Example: a(4) = 3; indeed, the Fibonacci cube Gamma(3) has the five vertices 000, 010, 001, 100, 101, three of which have an odd number of ones. See the E. Munarini et al. reference, p. 323. - Emeric Deutsch, Jun 28 2015
a(n) is the number of odd permutations p of 1..n such that |p(i)-i| <= 1 for i=1..n. - Dmitry Efimov, Jan 08 2016

T. D. Noe, Table of n, a(n) for n = 2..502
Shishuo Fu and Dazhao Tang, Partitions with fixed largest hook length, arXiv:1604.04028 [math.CO], 2016.
Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 886
S. Klavzar, Structure of Fibonacci cubes: a survey, J. Comb. Optim., 25, 2013, 505-522.
E. Munarini and N. Z. Salvi, Structural and enumerative properties of the Fibonacci cubes, Discrete Math., 255, 2002, 317-324.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1).

Cf. A000004, A000045, A001045, A005252, A010892, A113405, A191754.

[n le 4 select n else 2*Self(n-1)-Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 09 2016

LinearRecurrence[{2,-1,0,1},{1,2,3,4},39] (* Ray Chandler, Sep 23 2015 *)
CoefficientList[Series[1/((1-x-x^2)(1-x+x^2)), {x,0,40}], x] (* Vincenzo Librandi, Jan 09 2016 *)

Vec(x^2/((1-x-x^2)*(1-x+x^2)) + O(x^50)) \\ Michel Marcus, Feb 03 2016

def A024490(n): return (fibonacci(n+1) -chebyshev_U(n,1/2))/2
[A024490(n) for n in range(2,60)] # G. C. Greubel, Apr 10 2023

a(n) = A000045(n+1) - A005252(n).
a(n) = (A000045(n+1) - A010892(n))/2. - Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
a(n) = Sum_{k=0..n} Fibonacci(k+1)*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3). - Paul Barry, May 18 2004
G.f.: x^2/((1-x-x^2)(1-x+x^2)). - Jon Perry, Jun 22 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1,k+1)*(1+(-1)^k)/2. - Paul Barry, Jul 05 2007
G.f.: (1 + Q(0)*x^4/2)/(1-x)^2, where Q(k) = 1 + 1/(1 - x*( 4*k + 2 - x + x^3)/( x*(4*k + 4 - x + x^3) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 07 2014
E.g.f.: exp(x/2)*(15*cosh(sqrt(5)*x/2) - 5*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/30. - Stefano Spezia, Aug 03 2022

Additional comments from Henry Bottomley, Apr 07 2000
Corrected by Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
Further corrections from Hugo van der Sanden, Oct 05 2006

A192366 Denominators of a companion to the Bernoulli numbers.

Original entry on oeis.org

1, 2, 2, 3, 6, 15, 30, 35, 70, 105, 210, 1155, 2310, 5005, 10010, 15015, 30030, 255255, 510510, 1616615, 3233230, 969969, 1939938, 22309287, 44618574, 37182145, 74364290, 111546435, 223092870, 3234846615, 6469693230

Offset: 0

Paul Curtz, Jul 01 2011

For the numerators of the companion to the Bernoulli numbers and detailed information see A191754.

			The first rows of BC(n,m) matrix are
0,      1/2,   1/2,   1/3,     1/6,   1/15,
1/2,      0,  -1/6,  -1/6,   -1/10,  -1/30,
-1/2,  -1/6,     0,  1/15,    1/15,   1/35,
1/3,    1/6,  1/15,     0,  -4/105, -4/105,
-1/6, -1/10, -1/15, -4/105,      0,  4/105,
1/15,  1/30,  1/35,  4/105,  4/105,      0.

Cf. A191754 (numerator).

nmax:=30: mmax:=nmax: A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end: A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end: for m from 0 to 2*mmax do T(0,m) := A164555(m)/A027642(m) od: for n from 1 to nmax do for m from 0 to 2*mmax do T(n,m) := T(n-1,m+1)-T(n-1,m) od: od: for n from 0 to nmax do BC(n,n) :=0 : BC(n,n+1) := T(n,n+1) od: for m from 2 to 2*mmax do for n from 0 to m-2 do BC(n,m) := BC(n,m-1) + BC(n+1,m-1) od: od: for n from 0 to 2*nmax do BC(n,0) := (-1)^(n+1)*BC(0,n) od: for m from 1 to mmax do for n from 2 to 2*nmax do BC(n,m) := BC(n,m-1) + BC(n+1,m-1) od: od: for n from 0 to nmax do seq(BC(n,m),m=0..mmax) od: seq(BC(0,n),n=0..nmax): seq(denom(BC(0,n)), n=0..nmax); [Johannes W. Meijer, Jul 02 2011]

max = 30; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, max}]; diff = Table[ Differences[bb, n], {n, 1, Ceiling[max/2]}]; dd = Diagonal[diff]; bc[n_, n_] = 0; bc[n_, m_] /; m < n := bc[n, m] = bc[n-1, m+1] - bc[n-1, m]; bc[n_, m_] /; m == n+1 := bc[n, m] = -dd[[n+1]]; bc[n_, m_] /; m > n+1 := bc[n, m] = bc[n, m-1] + bc[n+1, m-1]; Table[bc[0, m], {m, 0, max}] // Denominator (* Jean-François Alcover, Aug 08 2012 *)

a(2*n+2)/a(2*n+1) = 2 for n>1.

Edited and Maple program added by Johannes W. Meijer, Jul 02 2011.

A191972 The numerators of T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0.

Original entry on oeis.org

1, -1, 1, -4, 4, -16, 3056, -1856, 181312, -35853056, 1670556928, -39832634368, 545273832448, -19385421824, 53026545299456, -2753673793480966144, 68423881271489019904, -22654998127210332160

Offset: 0

Paul Curtz, Jun 20 2011

For the denominators of T(n, n+1) see A190339, where detailed information can be found.

			T(n,n+1) = [1/2, -1/6, 1/15 , -4/105, 4/105, -16/231, 3056/15015, -1856/2145, 181312/36465, ...]

Cf. A191302, A191754, A181131.

nmax:=20: mmax:=nmax: A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end: A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end: for m from 0 to 2*mmax do T(0,m):=A164555(m)/A027642(m) od: for n from 1 to nmax do for m from 0 to 2*mmax do T(n,m):=T(n-1,m+1)-T(n-1,m) od: od: for n from 0 to nmax do seq(T(n,m),m=0..mmax) od: seq(numer(T(n,n+1)),n=0..nmax-1); # Johannes W. Meijer, Jun 30 2011

nmax = 17; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax+1}]; dd = Table[Differences[bb, n], {n, 1, nmax }]; a[0] = 1; a[n_] := dd[[n, n+2]] // Numerator; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 02 2012 *)

T(n, n+1) = T(n, n)/2.

a(n+2) = (-1)^n*A181130(n+2)/2.

Thanks to R. J. Mathar by Paul Curtz, Jun 20 2011

Edited by Johannes W. Meijer, Jun 30 2011

cp's OEIS Frontend

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

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A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2*m-1,2*m+1), where m = floor((n-2)/4).

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A192366 Denominators of a companion to the Bernoulli numbers.

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A191972 The numerators of T(n, n+1) with T(0, m) = A164555(m)/A027642(m) and T(n, m) = T(n-1, m+1) - T(n-1, m), n >= 1, m >= 0.

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

A024490 a(n) = C(n-1,1) + C(n-3,3) + ... + C(n-2m-1,2m+1), where m = floor((n-2)/4).