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.

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

Author

Paul Curtz, May 28 2016

Keywords

• nonn
• frac

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

• Seiichi Manyama, Table of n, a(n) for n = 0..3320
• A. F. Horadam, Oresme numbers, Fib. Quart., 12 (1974), 267-271.

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