A101925 -id:A101925 - cp's OEIS frontend

A089279 a(n) = 2 + sum(k=1 to n) [(-1)^k A001511(k)].

1, 3, 2, 5, 4, 6, 5, 9, 8, 10, 9, 12, 11, 13, 12, 17, 16, 18, 17, 20, 19, 21, 20, 24, 23, 25, 24, 27, 26, 28, 27, 33, 32, 34, 33, 36, 35, 37, 36, 40, 39, 41, 40, 43, 42, 44, 43, 48, 47, 49, 48, 51, 50, 52, 51, 55, 54, 56, 55, 58, 57, 59, 58, 65, 64, 66, 65, 68, 67, 69, 68, 72

Offset: 1

Gary W. Adamson, Oct 28 2003

nonn

a(2^n) = 2^n+1.

Cf. A001511, A005187, A101925.

a(2n+1) = A101925(n), a(2n+2) = a(2n+3)+1. - Ralf Stephan, Dec 28 2004

Edited by Don Reble, Nov 15 2005

A240988 Denominators of the (reduced) rationals (((n-1)!!)/(n!! * 2^((1 + (-1)^n)/2)))^((-1)^n), where n is a positive integer.

Original entry on oeis.org

1, 4, 2, 16, 8, 32, 16, 256, 128, 512, 256, 2048, 1024, 4096, 2048, 65536, 32768, 131072, 65536, 524288, 262144, 1048576, 524288, 8388608, 4194304, 16777216, 8388608, 67108864, 33554432, 134217728, 67108864, 4294967296, 2147483648, 8589934592, 4294967296

Offset: 1

James Burling, Aug 06 2014

Numerators for this sequence are the swinging factorial A163590, starting from n = 1.
The terms are all powers of 2 (A000079).
It appears that a(2*n) = 2^A101925(n) and a(2*n+1) = 2^A005187(n). - Robert Israel, Aug 06 2014

			For n = 1, a(1) = 1.
For n = 2, a(2) = 2 * 2 = 4.
For n = 6, a(6) = 2 * 2 * 4 * 2 = 32.

James Burling, The Special Rational Sequence

Cf. A163590 (numerators).

f:= n -> denom(((doublefactorial(n-1)) / (doublefactorial(n)*2^((1+(-1)^n)/2)))^((-1)^n)):
seq(f(n), n=1..100); # Robert Israel, Aug 06 2014

df(n) = prod(i=0, floor((n-1)/2), n-2*i) \\ Double factorial (n!!)
a(n) = denominator(((df(n-1)) / (df(n)*2^((1+(-1)^n)/2)))^((-1)^n))
vector(50, n, a(n)) \\ Colin Barker, Aug 06 2014

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

a(n) = denominator(g(1, n)) where g(m, n) = m if m = n; m/(2 * g(m + 1, n)) otherwise.

More terms from Colin Barker, Aug 06 2014

A361198 Consider a perfect infinite binary tree with nodes labeled with distinct positive integers where n appears at level A082850(n) and each level is filled from left to right; a(n) is the sibling of n in this tree.

Original entry on oeis.org

2, 1, 6, 5, 4, 3, 14, 9, 8, 13, 12, 11, 10, 7, 30, 17, 16, 21, 20, 19, 18, 29, 24, 23, 28, 27, 26, 25, 22, 15, 62, 33, 32, 37, 36, 35, 34, 45, 40, 39, 44, 43, 42, 41, 38, 61, 48, 47, 52, 51, 50, 49, 60, 55, 54, 59, 58, 57, 56, 53, 46, 31, 126, 65, 64, 69, 68

Offset: 1

Rémy Sigrist, Mar 04 2023

nonn

This sequence is a self-inverse permutation of the positive integers with no fixed point.
We can build a similar tree from any sequence of positive integers where each value appears infinitely many times. The choice of A082850 is interesting as each parent node appears immediately after its second child; also, for each pair of nodes of the same level, say p and p', and each pair of nodes, c and c', where c has ancestor p and c' has ancestor p', and the relative position of p with respect to c is the same as for p' with respect to c', we have p - c = p' - c'.
Empirically: to compute a(n): replace the least significant nonzero digit in the skew-binary expansion of n, say d, by 3-d. - Rémy Sigrist, Mar 02 2025

			The perfect infinite binary tree starts as follows:
                                 31
                  ---------------------------------
                 15                              30
          -----------------               -----------------
          7              14              22              29
      ---------       ---------       ---------       ---------
      3       6      10      13      18      21      25      28
    -----   -----   -----   -----   -----   -----   -----   -----
    1   2   4   5   8   9  11  12  16  17  19  20  23  24  26  27
.
So a(1) = 2 and a(2) = 1, a(4) = 5 and a(5) = 4, etc.,
   a(3) = 6 and a(6) = 3, a(10) = 13 and a(13) = 10, etc.,
   a(7) = 14 and a(14) = 7, a(22) = 29 and a(29) = 22,
   a(15) = 30 and a(30) = 15.

Cf. A082850, A101925, A126646, A308187.

a(n) = { my (n0 = n); for (h = 2, oo, if (n < 2^h-1, while (1, my (w=2^h-- - 1); if (n == w, return (n0 - n + 2*w), n == 2*w, return (n0 - n + w), n > w, n -= w)))) }

a(A101925(2*n)) = A101925(2*n + 1).
a(A101925(2*n + 1)) = A101925(2*n).
A082850(a(n)) = A082850(n).
a(2^k - 1) = 2^(k+1) - 2 for any k > 0.
abs(a(n) - n) = 2^A082850(n) - 1 (belongs to A126646).
Apparently, a(n) < n iff A308187(n+1) = 1.

cp's OEIS Frontend

A089279 a(n) = 2 + sum(k=1 to n) [(-1)^k A001511(k)].

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A240988 Denominators of the (reduced) rationals (((n-1)!!)/(n!! * 2^((1 + (-1)^n)/2)))^((-1)^n), where n is a positive integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A361198 Consider a perfect infinite binary tree with nodes labeled with distinct positive integers where n appears at level A082850(n) and each level is filled from left to right; a(n) is the sibling of n in this tree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula