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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A273692 a(n) is the denominator of 2*O(n+1) - O(n+2) where O(n) = n/2^n, the n-th Oresme number.

Original entry on oeis.org

2, 8, 2, 32, 32, 128, 64, 512, 512, 2048, 128, 8192, 8192, 32768, 16384, 131072, 131072, 524288, 131072, 2097152, 2097152, 8388608, 4194304, 33554432, 33554432, 134217728, 16777216, 536870912, 536870912, 2147483648, 1073741824, 8589934592, 8589934592, 34359738368
Offset: 0

Views

Author

Paul Curtz, May 28 2016

Keywords

Comments

O(n) is the Horadam notation.
O(n) or Oresme(n) = n/2^n = 0, 1/2, 1/2, 3/8, 1/4, ... . The positive Oresme numbers are O(n+1) = A000265(n+1)/A075101(n+1). See A209308. Consider Oco(n) = 2*O(n+1) - O(n+2) = 1/2, 5/8, 1/2, 11/32, 7/32, ... = A075677(n+1)/a(n). (See Coll(n) in A209308.)
Oco(n) = 1/2, 5/8, 1/2, 11/32, 7/32, 17/128, 5/64, 23/512, 13/512, 29/2048, 1/128, 35/8192, 19/8192, ... . Compare to (2+3*n)/2^(n+2).
Differences table of Oco(n):
1/2, 5/8, 1/2, 11/32, 7/32, 17/128, 5/64, ...
1/8, -1/8, -5/32, -1/8, -11/128, -7/128, ...
-1/4, -1/32, 1/32, 5/128, 1/32, ...
7/32, 1/16, 1/128, -1/128, ...
-5/32, -7/128, -1/64, ...
13/128, 5/128, ...
-1/16, ... .
First column: Io(n) = 1/2 followed by (-1)^n* A067745(n)/(8, 4, 32, 32, ...).
1) Alternated Oco(2n) + Io(2n) and Oco(2n+1) - Io(2n+1) gives 2^n.
2) Alternated Oco(2n) - Io(2n) and Oco(2n+1) + Io(2n+1) gives 3*O(n)/2.
(1/2 - 1/2 = 0, 5/8 + 1/8 = 3/4, 1/2 + 1/4 = 3/4, 11/32 + 7/32 = 9/16, ...)

References

  • M. R. Bacon and C. K. Cook, Some properties of Oresme numbers and convolutions ..., Fib. Q., 62:3 (2024), 233-240.

Links

Crossrefs

Cf. A000079, A000265, A000302, A004171 (main diagonal), A006519, A016789, A067745, A075101, A075677, A209308.

Programs

  • PARI
    Or(n) = n/2^n;
a(n) = denominator(2*Or(n+1) - Or(n+2)); \\ Michel Marcus, May 28 2016

Formula

a(n) = denominator of (2+3*n)/2^(n+2).
a(2n+1) = 8*4^n.
a(2n+2) = a(2n+1)/(4, 1, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 1, ..., shifted A006519?).

Extensions

More terms from Michel Marcus, May 28 2016

A273893 Denominator of n/3^n.

Original entry on oeis.org

1, 3, 9, 9, 81, 243, 243, 2187, 6561, 2187, 59049, 177147, 177147, 1594323, 4782969, 4782969, 43046721, 129140163, 43046721, 1162261467, 3486784401, 3486784401, 31381059609, 94143178827, 94143178827, 847288609443, 2541865828329, 282429536481, 22876792454961
Offset: 0

Views

Author

Paul Curtz, Jun 02 2016

Keywords

Comments

The reduced values are Ms(n) = 0, 1/3, 2/9, 1/9, 4/81, 5/243, 2/243, 7/2187, 8/6561, 1/2187, ... .
Numerators: 0, 1, 2, 1, 4, ... = A038502(n).
Ms(-n) = 0, -3, -18, ... = - A036290(n).
Difference table of Ms(n):
0, 1/3, 2/9, 1/9, 4/81, 5/243, 2/243, ...
1/3, -1/9, -1/9, -5/81, -7/243, -1/81, ...
-4/9, 0, 4/81, 8/243, 4/243, ...
4/9, 4/81, -4/243, -4/243, ...
-32/81, -16/243, 0, ...
80/243, 16/243, ...
-64/243, ...
etc.
The difference table of O(n) = n/2^n (Oresme numbers) has its 0's on the main diagonal. Here the 0's appear every two rows. For n/4^n,they appear every three rows. (The denominators of O(n) are 2^A093048(n)).
All terms are powers of 3 (A000244).

Crossrefs

Cf. A007949, A000244, A013732, A013733, A036290, A038502, A075101.

Programs

  • Mathematica
    Table[Denominator[n/3^n], {n, 0, 28}] (* Michael De Vlieger, Jun 03 2016 *)
  • PARI
    a(n) = denominator(n/3^n) \\ Felix Fröhlich, Jun 07 2016
  • Sage
    [1] + [3^(n-n.valuation(3)) for n in [1..30]] # Tom Edgar, Jun 02 2016

Formula

For n>0, a(n) = 3^(n - valuation(n,3)) = 3^(n - A007949(n)). - Tom Edgar, Jun 02 2016
a(3n+1) = 3^(3n+1), a(3n+2) = 3^(3n+2).
a(3n+6) = 27*(3n+3).
From Peter Bala, Feb 25 2019: (Start)
a(n) = 3^n/gcd(n,3^n).
O.g.f.: 1 + F(3*x) - (2/3)*F((3*x)^3) - (2/9)*F((3*x)^9) - (2/27)*F((3*x)^27) - ..., where F(x) = x/(1 - x).
O.g.f. for reciprocals: Sum_{n >= 0} x^n/a(n) = 1 + F((x/3)) + 2*( F((x/3)^3) + 3*F((x/3)^9) + 9*F((x/3)^27) + ... ). Cf. A038502. (End)

A279635 Denominator of (0 followed by A005126(n)= 2, 4, 7, ...)/2^n, a sequence corresponding to A271573.

Original entry on oeis.org

1, 1, 1, 8, 4, 32, 32, 128, 32, 512, 512, 2048, 1024, 8192, 8192, 32768, 4096, 131072, 131072, 524288, 262144, 2097152, 2097152, 8388608, 2097152, 33554432, 33554432, 134217728, 67108864, 536870912, 536870912, 2147483648, 134217728, 8589934592, 8589934592, 34359738368, 17179869184, 137438953472, 137438953472, 549755813888, 137438953472
Offset: 0

Views

Author

Jean-François Alcover and Paul Curtz, Dec 16 2016

Keywords

Crossrefs

Cf. A005126, A075101, A271573.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Denominator[(2^(n-1)+n)/2^n]; Table[a[n], {n, 0, 40}]
(* or *)
a[0] = 1; a[n_] := 2^(n-IntegerExponent[2^(n-1)+n, 2]); Table[a[n], {n, 0, 40}]

Formula

a(n) = 2^(n-valuation(2^(n-1)+n,2)), with a(0) = 1.

A273153 a(n) = Numerator of (0 followed by 1's) - n/2^n.

Original entry on oeis.org

0, 1, 1, 5, 3, 27, 29, 121, 31, 503, 507, 2037, 1021, 8179, 8185, 32753, 4095, 131055, 131063, 524269, 262139, 2097131, 2097141, 8388585, 2097149, 33554407, 33554419, 134217701, 67108857, 536870883, 536870897, 2147483617, 134217727, 8589934559, 8589934575, 34359738333
Offset: 0

Views

Author

Paul Curtz, May 16 2016

Keywords

Comments

A060576(n+1) = 0, 1, 1, 1, 1, 1, 1, ... - (0(n) = Oresme(n) = 0, 1/2, 1/2, 3/8, 1/4, 5/32, 3/32, ...). Both sequences are autosequences of the first kind. f(n) = 0, 1/2, 1/2, 5/8, 3/4, 27/32, 29/32, 121/128, ... is an autosequence of the first kind. Without one 1/2, f(n) is an increasing sequence.
The numerators (1 followed by A075101(n)) are the same as in n/2^n.

Examples

			Array of differences of fractions (characteristic aspect of an autosequence of the first kind):
0,     1/2,   1/2,   5/8,   3/4, ...
1/2,     0,   1/8,   1/8,  3/32, ...
-1/2,  1/8,     0, -1/32, -1/32, ...
5/8,  -1/8, -1/32,     0, 1/128, ...
-3/4, 3/32,  1/32, 1/128,     0, ...
...

Links

Crossrefs

Cf. A060576, A054977, A075101, A209308.

Programs

  • Mathematica
    {0}~Join~Array[Numerator@ Abs[1 - Binomial[0, # - 1] - #/2^#] &, 30] (* Michael De Vlieger, May 17 2016 *)
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