cp's OEIS Frontend

A324338 a(n) = A002487(1+A006068(n)).

%I A324338 #30 Jan 12 2020 13:50:18
%S A324338 1,1,1,2,1,3,3,2,1,4,5,3,4,3,2,5,1,5,7,4,7,5,3,8,5,4,3,7,2,7,8,5,1,6,
%T A324338 9,5,10,7,4,11,9,7,5,12,3,11,13,8,6,5,4,9,3,10,11,7,2,9,12,7,11,8,5,
%U A324338 13,1,7,11,6,13,9,5,14,13,10,7,17,4,15,18,11,11,9,7,16,5,17,19,12,3,14,19,11,18,13,8,21,7,6,5,11,4,13,14,9,3,13
%N A324338 a(n) = A002487(1+A006068(n)).
%C A324338 Like in A324337, a few terms preceding each 2^k-th term (here always 1) seem to consist of a batch of nearby Fibonacci numbers (A000045) in some order. For example, a(65533) = 987, a(65534) = 610 and a(65535) = 1597.
%C A324338 For all n > 0 A324338(n)/A324337(n) constitutes an enumeration system of all positive rationals. For all n > 0 A324338(n) + A324337(n) = A071585(n). - _Yosu Yurramendi_, Oct 22 2019
%H A324338 Antti Karttunen, <a href="/A324338/b324338.txt">Table of n, a(n) for n = 0..16384</a>
%H A324338 Antti Karttunen, <a href="/A324338/a324338.txt">Data supplement: n, a(n) computed for n = 0..65537</a>
%H A324338 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A324338 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F A324338 a(n) = A002487(1+A006068(n)).
%F A324338 a(2^n) = 1 for all n >= 0.
%F A324338 From _Yosu Yurramendi_, Oct 22 2019: (Start)
%F A324338 a(2^m+2^(m-1)+k) = A324337(2^m+        k), m > 0, 0 <= k < 2^(m-1)
%F A324338 a(2^m+        k) = A324337(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1). (End)
%F A324338 a(n) = A324337(A063946(n)), n > 0. _Yosu Yurramendi_, Nov 04 2019
%F A324338 a(n) = A002487(A233279(n)), n > 0. _Yosu Yurramendi_, Nov 08 2019
%F A324338 From _Yosu Yurramendi_, Nov 28 2019: (Start)
%F A324338 a(2^(m+1)+k) - a(2^m+k) = A324337(k), m >= 0,  0 <= k < 2^m.
%F A324338 a(A059893(2^(m+1)+A000069(k+1))) - a(A059893(2^m+A000069(k+1))) =  A071585(k), m >= 1,  0 <= k < 2^(m-1).
%F A324338 a(A059893(2^m+ A001969(k+1))) = A071585(k),    m >= 0,  0 <= k < 2^(m-1). (End)
%F A324338 From _Yosu Yurramendi_, Nov 29 2019: (Start)
%F A324338 For n > 0:
%F A324338 A324338(n) + A324337(n) = A071585(n).
%F A324338 A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
%F A324338 A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
%F A324338 A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
%F A324338 A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
%F A324338 a(n) = A002487(A233279(n)). _Yosu Yurramendi_, Dec 27 2019
%o A324338 (PARI)
%o A324338 A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
%o A324338 A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); };
%o A324338 A324338(n) = A002487(1+A006068(n));
%o A324338 (R)
%o A324338 maxlevel <- 6 # by choice #
%o A324338 b <- 0; A324338 <- 1; A324337 <- 1
%o A324338 for(i in 1:2^maxlevel) {
%o A324338   b[2*i  ] <-     b[i]
%o A324338   b[2*i+1] <- 1 - b[i]
%o A324338   A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
%o A324338   A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
%o A324338   A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
%o A324338   A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
%o A324338 #
%o A324338 A324338[1:127]; A324337[1:127]
%o A324338 # _Yosu Yurramendi_, Oct 22 2019
%Y A324338 Cf. A000045, A002487, A006068, A324288, A324337.
%K A324338 nonn
%O A324338 0,4
%A A324338 _Antti Karttunen_, Feb 23 2019