A273493 a(n) = A245327(n) + A245328(n).

2, 3, 3, 5, 5, 4, 4, 8, 8, 7, 7, 7, 7, 5, 5, 13, 13, 11, 11, 12, 12, 9, 9, 11, 11, 10, 10, 9, 9, 6, 6, 21, 21, 18, 18, 19, 19, 14, 14, 19, 19, 17, 17, 16, 16, 11, 11, 18, 18, 15, 15, 17, 17, 13, 13, 14, 14, 13, 13, 11, 11, 7, 7, 34, 34, 29, 29, 31, 31, 23, 23, 30, 30, 27, 27, 25, 25, 17, 17, 31, 31, 26, 26, 29, 29, 22, 22, 25

Offset: 1

Author

Yosu Yurramendi, May 23 2016

Keywords

• nonn

Comments

The terms (n>0) may be written as a left-justified array with rows of length 2^m, m >= 0:

2,

3, 3,

5, 5, 4, 4,

8, 8, 7, 7, 7, 7, 5, 5,

13,13,11,11,12,12, 9, 9,11,11,10,10, 9, 9, 6, 6,

21,21,18,18,19,19,14,14,19,19,17,17,16,16,11,11,18,18,15,15,17,17,13,13,14,14,...

All columns have the Fibonacci sequence property: a(2^(m+2) + k) = a(2^(m+1) + k) + a(2^m + k), m >= 0, 0 <= k < 2^m (empirical observations).

The terms (n>0) may also be written as a right-justified array with rows of length 2^m, m >= 0:

2,

3, 3,

5, 5, 4, 4,

8, 8, 7, 7, 7, 7, 5, 5,

13,13,11,11,12,12, 9, 9,11,11,10,10, 9, 9, 6, 6,

...,19,19,17,17,16,16,11,11,18,18,15,15,17,17,13,13,14,14,13,13,11,11, 7, 7,

Each column is an arithmetic sequence. The differences of the arithmetic sequences repeat the sequence A071585: a(2^(m+2) -1 - 2k) - a(2^(m+1) -1 - 2k) = A071585(k-1), m > 0, 0 <= k < 2^m ; a(2^(m+2) -1 - 2k - 1) - a(2^(m+1) -1 - 2k - 1) = A071585(k-1), m > 0, 0 <= k < 2^m .

n>1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (Franklin T. Adams-Watters' comment), that is the sequence obtained by adding numerator and denominator in the Calkin-Wilf enumeration system of positive rationals. A245327(n)/A245328(n) is also an enumeration system of all positive rationals, and in each level m >= 0 (ranks between 2^m and 2^(m+1)-1) rationals are the same in both systems. Thus a(n) has the same terms in each level as A007306.

a(n) = A273494(A059893(n)), a(A059893(n)) = A273494(n), n > 0. - Yosu Yurramendi, May 30 2017

Links

• Yosu Yurramendi, Table of n, a(n) for n = 1..4099

Crossrefs

Cf. A007306, A268087, A071585, A086592, A273494

Programs

PARI

b(n) = my(b=binary(n)); fromdigits(concat(b[1], Vecrev(vector(#b-1, k, b[k+1]))), 2); \\ from A059893
a(n) = my(n=b(n), x=1, y=1); for(i=0, logint(n, 2), if(bittest(n, i), [x, y]=[x+y, y], [x, y]=[y, x+y])); x \\ Mikhail Kurkov, Mar 11 2023

Formula

a(n) = A007306(A284459(n)), n > 0. - Yosu Yurramendi, Aug 23 2021