This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A273494 #42 Apr 25 2024 23:59:55 %S A273494 2,3,3,5,4,5,4,8,7,7,5,8,7,7,5,13,11,12,9,11,10,9,6,13,11,12,9,11,10, %T A273494 9,6,21,18,19,14,19,17,16,11,18,15,17,13,14,13,11,7,21,18,19,14,19,17, %U A273494 16,11,18,15,17,13,14,13,11,7,34,29,31,23,30,27,25,17,31,26,29,22,25,23,20,13,29,25,26,19,27,24,23,16,23 %N A273494 a(n) = A245325(n) + A245326(n). %C A273494 The terms (n>0) may be written as a left-justified array with rows of length 2^m, m >= 0: %C A273494 2, %C A273494 3, 3, %C A273494 5, 4, 5, 4, %C A273494 8, 7, 7, 5, 8, 7, 7, 5, %C A273494 13,11,12, 9,11,10, 9, 6,13,11,12, 9,11,10, 9, 6, %C A273494 21,18,19,14,19,17,16,11,18,15,17,13,14,13,11, 7,21,18,19,14,19,17,... %C A273494 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 A273494 The terms (n>0) may also be written as a right-justified array with rows of length 2^m, m >= 0: %C A273494 2, %C A273494 3, 3, %C A273494 5, 4, 5, 4, %C A273494 8, 7, 7, 5, 8, 7, 7, 5, %C A273494 13,11,12, 9,11,10, 9, 6,13,11,12, 9,11,10, 9, 6, %C A273494 ..., 18,15,17,13,14,13,11, 7,21,18,19,14,19,17,16,11,18,15,17,13,14,13,11, 7, %C A273494 Each column is an arithmetic sequence. The differences of the arithmetic sequences give the sequence A071585: a(2^(m+1)-1-k) - a(2^m-1-k) = A071585(k), m >= 0, 0 <= k < 2^m. %C A273494 n > 1 occurs in this sequence phi(n) = A000010(n) times, as it occurs in A007306 (_Franklin T. Adams-Watters_'s comment), which is the sequence obtained by adding numerator and denominator in the Calkin-Wilf enumeration system of positive rationals. A245325(n)/A245326(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 A273494 The same property occurs in all numerator+denominator sequences of enumeration systems of positive rationals, as, for example, A007306 (A007305+A047679), A071585 (A229742+A071766), A086592 (A020650+A020651), A268087 (A162909+A162910). %H A273494 Yosu Yurramendi, <a href="/A273494/b273494.txt">Table of n, a(n) for n = 1..4095</a> %F A273494 a(n) = A273493(A059893(n)), a(A059893(n)) = A273493(n), n > 0. - _Yosu Yurramendi_, May 30 2017 %F A273494 a(n) = A007306(A059893(A180200(n))) = A007306(A059894(A154435(n))). - _Yosu Yurramendi_, Sep 20 2021 %o A273494 (PARI) a(n) = my(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 10 2023 %Y A273494 Cf. A007306, A268087, A071585, A086592, A273493 %K A273494 nonn %O A273494 1,1 %A A273494 _Yosu Yurramendi_, May 23 2016