cp's OEIS Frontend

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

%I A273493 #34 Apr 23 2024 22:36:13
%S A273493 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,
%T A273493 6,6,21,21,18,18,19,19,14,14,19,19,17,17,16,16,11,11,18,18,15,15,17,
%U A273493 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
%N A273493 a(n) = A245327(n) + A245328(n).
%C A273493 The terms (n>0) may be written as a left-justified array with rows of length 2^m, m >= 0:
%C A273493 2,
%C A273493 3, 3,
%C A273493 5, 5, 4, 4,
%C A273493 8, 8, 7, 7, 7, 7, 5, 5,
%C A273493 13,13,11,11,12,12, 9, 9,11,11,10,10, 9, 9, 6, 6,
%C A273493 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,...
%C A273493 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).
%C A273493 The terms (n>0) may also be written as a right-justified array with rows of length 2^m, m >= 0:
%C A273493                                                                               2,
%C A273493                                                                            3, 3,
%C A273493                                                                      5, 5, 4, 4,
%C A273493                                                          8, 8, 7, 7, 7, 7, 5, 5,
%C A273493                                 13,13,11,11,12,12, 9, 9,11,11,10,10, 9, 9, 6, 6,
%C A273493     ...,19,19,17,17,16,16,11,11,18,18,15,15,17,17,13,13,14,14,13,13,11,11, 7, 7,
%C A273493 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 .
%C A273493 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.
%C A273493 a(n) = A273494(A059893(n)), a(A059893(n)) = A273494(n), n > 0. - _Yosu Yurramendi_, May 30 2017
%H A273493 Yosu Yurramendi, <a href="/A273493/b273493.txt">Table of n, a(n) for n = 1..4099</a>
%F A273493 a(n) = A007306(A284459(n)), n > 0. - _Yosu Yurramendi_, Aug 23 2021
%o A273493 (PARI) b(n) = my(b=binary(n)); fromdigits(concat(b[1], Vecrev(vector(#b-1, k, b[k+1]))), 2); \\ from A059893
%o A273493 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
%Y A273493 Cf. A007306, A268087, A071585, A086592, A273494
%K A273493 nonn
%O A273493 1,1
%A A273493 _Yosu Yurramendi_, May 23 2016