cp's OEIS Frontend

A324337 a(n) = A002487(A006068(n)).

%I A324337 #45 Mar 19 2025 08:25:21
%S A324337 0,1,2,1,3,2,1,3,4,3,2,5,1,4,5,3,5,4,3,7,2,7,8,5,1,5,7,4,7,5,3,8,6,5,
%T A324337 4,9,3,10,11,7,2,9,12,7,11,8,5,13,1,6,9,5,10,7,4,11,9,7,5,12,3,11,13,
%U A324337 8,7,6,5,11,4,13,14,9,3,13,17,10,15,11,7,18,2,11,16,9,17,12,7,19,14,11,8,19,5,18,21,13,1,7,11,6,13,9,5,14,13,10
%N A324337 a(n) = A002487(A006068(n)).
%C A324337 Like in A324338, a few terms preceding each position n = 2^k seem to be a batch of nearby Fibonacci numbers in some order.
%C A324337 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 A324337 Antti Karttunen, <a href="/A324337/b324337.txt">Table of n, a(n) for n = 0..16384</a>
%H A324337 Antti Karttunen, <a href="/A324337/a324337.txt">Data supplement: n, a(n) computed for n = 0..65537</a>
%H A324337 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A324337 <a href="/index/St#Stern">Index entries for sequences related to Stern's sequences</a>
%F A324337 From _Yosu Yurramendi_, Oct 22 2019: (Start)
%F A324337 a(2^m+        k) = A324338(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1)
%F A324337 a(2^m+2^(m-1)+k) = A324338(2^m+        k), m > 0, 0 <= k < 2^(m-1). (End)
%F A324337 a(n) = A324338(A063946(n)), n > 0. _Yosu Yurramendi_, Nov 04 2019
%F A324337 a(n) = A002487(A248663(A283477(n))). - _Antti Karttunen_, Nov 06 2019
%F A324337 a(n) = A002487(1+A233279(n)). - _Yosu Yurramendi_, Nov 08 2019
%F A324337 From _Yosu Yurramendi_, Nov 28 2019: (Start)
%F A324337 a(2^(m+1)+k) - a(2^m+k) = A324338(k), m >= 0, 0 <= k < 2^m.
%F A324337 a(A059893(2^(m+1)+A001969(k+1))) - a(A059893(2^m+A001969(k+1))) = A071585(k), m >= 0, 0 <= k < 2^(m-1).
%F A324337 a(A059893(2^(m+1)+ A000069(k+1))) = A071585(k), m >= 1, 0 <= k < 2^(m-1). (End)
%F A324337 From _Yosu Yurramendi_, Nov 29 2019: (Start)
%F A324337 For n > 0:
%F A324337 A324338(n) + A324337(n) = A071585(n).
%F A324337 A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)
%F A324337 A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)
%F A324337 A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)
%F A324337 A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)
%F A324337 a(n) = A002487(1+A233279(n)). - _Yosu Yurramendi_, Dec 27 2019
%t A324337 Block[{f}, f[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b = a + b, a = a + b]; n = Floor[n/2]]; b]; Array[f@ Fold[BitXor, #, Quotient[#, 2^Range[BitLength[#] - 1]]] &, 106, 0]] (* _Michael De Vlieger_, Dec 14 2019, after _Jean-François Alcover_ at A002487 and _Jan Mangaldan_ at A006068 *)
%o A324337 (PARI)
%o A324337 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 A324337 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 A324337 A324337(n) = A002487(A006068(n));
%o A324337 (R)
%o A324337 maxlevel <- 6 # by choice
%o A324337 #
%o A324337 b <- 0; A324338 <- 1; A324337 <- 1
%o A324337 for(i in 1:2^maxlevel) {
%o A324337   b[2*i  ] <-     b[i]
%o A324337   b[2*i+1] <- 1 - b[i]
%o A324337   A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]
%o A324337   A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])
%o A324337   A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]
%o A324337   A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}
%o A324337 #
%o A324337 A324338[1:127]; A324337[1:127]
%o A324337 # _Yosu Yurramendi_, Oct 22 2019
%Y A324337 Cf. A002487, A006068, A063946, A071585, A248663, A283477, A324287, A324338.
%K A324337 nonn
%O A324337 0,3
%A A324337 _Antti Karttunen_, Feb 23 2019