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

0,1

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.

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?).

Prog

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

Crossrefs

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

Sequence in context: A245995 A085192 A320972 * A006085 A376265 A021357

Adjacent sequences: A273689 A273690 A273691 * A273693 A273694 A273695

Keyword

nonn,frac

Author

Paul Curtz, May 28 2016

Extensions

More terms from Michel Marcus, May 28 2016

Status

approved