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.

Showing 1-7 of 7 results.

A194662 Number of k in [1,n] for which + > 1, where < > = fractional part and r = (1+sqrt(5))/2 (the golden ratio); row sums of A164661.

Original entry on oeis.org

1, 0, 3, 2, 0, 4, 2, 8, 6, 2, 9, 5, 0, 9, 3, 14, 9, 2, 14, 7, 21, 14, 5, 21, 12, 2, 19, 8, 27, 17, 5, 25, 13, 0, 22, 8, 32, 18, 3, 28, 13, 40, 25, 8, 37, 20, 2, 32, 13, 45, 27, 7, 40, 20, 55, 35, 13, 50, 28, 5, 43, 20, 60, 37, 12, 53, 28, 2, 45, 18, 63, 36, 8, 54, 26, 74, 46
Offset: 1

Views

Author

Clark Kimberling, Sep 01 2011

Keywords

Crossrefs

Cf. A194661.

Programs

  • Mathematica
    r = GoldenRatio; z = 14;
p[x_] := FractionalPart[x]; f[x_] := Floor[x];
h[n_, k_] := f[p[n*r] + p[k*r]]
Flatten[Table[h[n, k], {n, 1, z}, {k, 1, n}]]  (* A194661 *)
TableForm[Table[h[n, k], {n, 1, z}, {k, 1, n}]]
t[n_] := Sum[h[n, k], {k, 1, n}]
Table[t[n], {n, 1, 100}]   (* A194662 *)

A164658 Numerators of coefficients of integrated Chebyshev polynomials T(n,x) (in increasing order of powers of x).

Original entry on oeis.org

1, 0, 1, -1, 0, 2, 0, -3, 0, 1, 1, 0, -8, 0, 8, 0, 5, 0, -5, 0, 8, -1, 0, 6, 0, -48, 0, 32, 0, -7, 0, 14, 0, -56, 0, 8, 1, 0, -32, 0, 32, 0, -256, 0, 128, 0, 9, 0, -30, 0, 72, 0, -72, 0, 128, -1, 0, 50, 0, -80, 0, 160, 0, -1280, 0, 512, 0, -11, 0, 55, 0, -616, 0, 352, 0, -1408, 0, 256, 1, 0, -24, 0, 168, 0, -512, 0, 768
Offset: 0

Views

Author

Wolfdieter Lang, Oct 16 2009

Keywords

Comments

The denominators are given in A164659.
The column m of the rational triangle A164658/A164659 when multiplied by m/2^(m-2) becomes (with shifted offset) the column no. m-1 divided by 2^(m-1) of the Chebyshev T-triangle A053120 for m=1,2,3,...

Examples

			Rationals a(n,m)/A164659(n,m) = [1], [0, 1/2], [-1, 0, 2/3], [0, -3/2, 0, 1], [1, 0, -8/3, 0, 8/5],...

Links

Crossrefs

Row sums of triangle give A164662.
A053120: coefficients of T-polynomials.
Row sums of rational triangle A164658/A164659 are given by A164660/A164661.

Programs

  • Mathematica
    row[n_] := CoefficientList[Integrate[ChebyshevT[n, x], x], x] // Rest // Numerator; Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 06 2016 *)

Formula

a(n,m) = numerator(b(n,m)), with int(T(n,x))= sum(b(n,m)*x^m,m=1..n+1), n>=0, where T(n,x) are Chebyshevs polynomials of the first kind.

A164659 Denominators of coefficients of integrated Chebyshev polynomials T(n,x) (in increasing order of powers of x).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 1, 3, 1, 5, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 7, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 7, 1, 9, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 9, 1, 11, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 13, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1
Offset: 0

Views

Author

Wolfdieter Lang, Oct 16 2009

Keywords

Comments

The numerators are given in A164658.
See the W. Lang link in A164658 for this table and the rational table A164658/A164659.

Examples

			Rational table A164658(n,m)/a(n,m) = [1], [0, 1/2], [-1, 0, 2/3], [0, -3/2, 0, 1], [1, 0, -8/3, 0, 8/5],...

Links

Crossrefs

Row sums of this triangle give A164663.
Row sums of rational triangle A164658/A164659 are given in A164660/A164661.

Programs

  • Mathematica
    row[n_] := CoefficientList[Integrate[ChebyshevT[n, x], x], x] // Rest // Denominator; Table[row[n], {n, 0, 13}] // Flatten (* Jean-François Alcover, Oct 06 2016 *)

Formula

a(n,m) = denominator(b(n,m)), with int(T(n,x),x)= sum(b(n,m)*x^m,m=1..n+1), n>=0, where T(n,x) are Chebyshevs polynomials of the first kind.

A164660 Numerators of row sums of triangle of rationals A164658/A164659. Definite integral of Chebyshev polynomials of the first kind: Integral_{x=0..1} T(n,x).

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
Offset: 0

Views

Author

Wolfdieter Lang, Oct 16 2009

Keywords

Examples

			Rationals a(n)/A164661(n)= [1, 1/2, -1/3, -1/2, -1/15, 1/6, -1/35, -1/6, -1/63, 1/10, -1/99, ...].

Links

Crossrefs

The denominators are given in A164661.
Triangle of int(T(n,x),x) coefficients is A164658/A164659.

Formula

a(n) = numerator(Sum_{m=1..n+1} IT(n,m)), n>=0, with IT(n,m):= A164658(n,m)/A164659(n,m) (coefficient triangle from the indefinite integral Integral_{x} T(n,x), n>=0, in lowest terms).
Conjecture for the rationals r(n):= A164660(n)/A164661(n): r(n)= 1 if n=0, if n is even r(n) = -1/((n-1)*(n+1)) and if n is odd r(n) = ((-1)^((n-1)/2))/(2*(2*floor((n-1)/4)+1)).
a(n+1) = Product_{k=1..n} ( 1-2*(floor(k^n/n)-floor((k^n -1)/n)) ) = (-1)^(A003557(n)) for n>0 (conjecture). - Anthony Browne, May 29 2016

A174111 Denominators of the image of a modified Bernoulli-number sequence under the Akiyama-Tanigawa transform.

Original entry on oeis.org

2, 3, 2, 15, 6, 7, 6, 15, 10, 33, 6, 455, 210, 3, 2, 255, 30, 133, 42, 33, 110, 69, 6, 455, 546, 3, 2, 435, 30, 2387, 462
Offset: 0

Views

Author

Paul Curtz, Mar 08 2010

Keywords

Comments

The image of the sequence A164555(k)/A027642(k), k>=0, under the Takiyama-Tanigawa
transform is
1/2, 2/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33, 5/6, 1382/455, -691/210, -49/3, 35/2, 28936/255, -3617/30, -131601/133, 43867/42, 349222/33, ...
The current sequence contains the denominators of this image.

Links

Crossrefs

Cf. A174110 (numerators), A164661.

Programs

  • Maple
    read("transforms3") ;
A174111 := proc(n) Lin := [bernoulli(0),-bernoulli(1),seq(bernoulli(k),k=2..n+1)] ; AKIYATANI(Lin) ; denom(op(n+1,%)) ; end proc:
  • Mathematica
    b[0]=0; b[1]=1; b[2]=1/2; b[n_] := BernoulliB[n-1]; a[0, m_] := b[m+1]; a[n_, m_] := a[n, m] = (m+1)*(a[n-1, m] - a[n-1, m+1]); Table[a[1, m], {m, 0, 30}] // Denominator  (* Jean-François Alcover, Aug 09 2012 *)

A164662 Row sums of triangle A164658 (numerators of coefficients from Integral_{x} T(n,x), with T(n,x) Chebyshev polynomials of the first kind).

Original entry on oeis.org

1, 1, 1, -2, 1, 8, -11, -41, -127, 107, -639, -1372, -3695, 514, -25983, -26339, -70655, -46299, -430955, -484134, -2808479, 93148, -5032895, -17319181, -72165695, 43371103, -171203135, -378398576, -148383647, -2605023034, -3368133419, 11479942073, -11902375935, 2021161097, -708801692671
Offset: 0

Views

Author

Wolfdieter Lang, Oct 16 2009

Keywords

Comments

The row sums of the rational triangle A164658/A164659 give A164660/A164661.

Links

Crossrefs

Row sums of denominator triangle A164659 gives A164663.

Formula

a(n) = Sum_{m=1..n+1} A164658(n,m), n>=0.

A164663 Row sums of triangle A164659 (denominators of coefficients from int(T(n,x),x), with T(n,x) Chebyshev polynomials of the first kind).

Original entry on oeis.org

1, 3, 5, 5, 11, 9, 17, 11, 25, 15, 31, 21, 35, 25, 49, 23, 55, 29, 53, 39, 71, 41, 77, 43, 79, 47, 97, 53, 95, 73, 89, 63, 115, 51, 165, 73, 131, 77, 135, 71, 141, 103, 125, 105, 163, 89, 169, 81, 161, 101, 193, 109, 181, 119, 197, 121, 261, 111, 203, 139, 179, 143, 239, 95
Offset: 0

Views

Author

Wolfdieter Lang, Oct 16 2009

Keywords

Comments

The row sums of the rational triangle A164658/A164659 give A164660/A164661.

Links

Crossrefs

Row sums of numerator triangle A164658 gives A164662.

Formula

a(n) = Sum_{m=1..n+1} A164659(n,m), n>=0.
Showing 1-7 of 7 results.

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