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A324338 a(n) = A002487(1+A006068(n)). 4
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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.

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Index entries for sequences related to binary expansion of n

Index entries for sequences related to Stern's sequences

FORMULA

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

a(2^n) = 1 for all n >= 0.

From Yosu Yurramendi, Oct 22 2019: (Start)

a(2^m+2^(m-1)+k) = A324337(2^m+        k), m > 0, 0 <= k < 2^(m-1)

a(2^m+        k) = A324337(2^m+2^(m-1)+k), m > 0, 0 <= k < 2^(m-1). (End)

a(n) = A324337(A063946(n)), n > 0. Yosu Yurramendi, Nov 04 2019

a(n) = A002487(A233279(n)), n > 0. Yosu Yurramendi, Nov 08 2019

From Yosu Yurramendi, Nov 28 2019: (Start)

a(2^(m+1)+k) - a(2^m+k) = A324337(k), m >= 0,  0 <= k < 2^m.

a(A059893(2^(m+1)+A000069(k+1))) - a(A059893(2^m+A000069(k+1))) =  A071585(k), m >= 1,  0 <= k < 2^(m-1).

a(A059893(2^m+ A001969(k+1))) = A071585(k),    m >= 0,  0 <= k < 2^(m-1). (End)

From Yosu Yurramendi, Nov 29 2019: (Start)

For n > 0:

A324338(n) + A324337(n) = A071585(n).

A324338(2*A001969(n)  )-A324337(2*A001969(n)  ) =  A071585(n-1)

A324338(2*A001969(n)+1)-A324337(2*A001969(n)+1) = -A324337(n-1)

A324338(2*A000069(n)  )-A324337(2*A000069(n)  ) = -A071585(n-1)

A324338(2*A000069(n)+1)-A324337(2*A000069(n)+1) =  A324338(n-1) (End)

a(n) = A002487(A233279(n)). Yosu Yurramendi, Dec 27 2019

PROG

(PARI)

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

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); };

A324338(n) = A002487(1+A006068(n));

(R)

maxlevel <- 6 # by choice #

b <- 0; A324338 <- 1; A324337 <- 1

for(i in 1:2^maxlevel) {

  b[2*i  ] <-     b[i]

  b[2*i+1] <- 1 - b[i]

  A324338[2*i  ] <- A324338[i]          + A324337[i]*   b[i]

  A324338[2*i+1] <- A324338[i]          + A324337[i]*(1-b[i])

  A324337[2*i  ] <- A324338[i]*(1-b[i]) + A324337[i]

  A324337[2*i+1] <- A324338[i]*   b[i]  + A324337[i]}

#

A324338[1:127]; A324337[1:127]

# Yosu Yurramendi, Oct 22 2019

CROSSREFS

Cf. A000045, A002487, A006068, A324288, A324337.

Sequence in context: A336692 A332434 A262209 * A047679 A179480 A245326

Adjacent sequences:  A324335 A324336 A324337 * A324339 A324340 A324341

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 23 2019

STATUS

approved

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Last modified June 17 00:17 EDT 2021. Contains 345080 sequences. (Running on oeis4.)