A257805 -id:A257805 - cp's OEIS frontend

A257806 a(n) = A257808(n) - A257807(n).

0, -1, 0, 1, 0, 1, 2, 1, 2, 1, 2, 3, 2, 3, 4, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 7, 6, 7, 8, 7, 6, 7, 8, 9, 10, 11, 12, 11, 12, 13, 12, 11, 10, 9, 10, 9, 10, 11, 10, 11, 12, 13, 12, 11, 12, 13, 12, 13, 12, 13, 14, 13, 12, 11, 10, 9, 10, 11, 12, 11, 10, 9, 10, 11, 12, 13, 14, 15, 14, 15, 16, 15, 16, 15, 14

Offset: 0

Antti Karttunen, May 12 2015

Alternative description: Start with a(0) = 0, and then to obtain each a(n), look at each successive term in the infinite trunk of inverted binary beanstalk, from A233271(1) onward, subtracting one from a(n-1) if A233271(n) is odd, and adding one to a(n-1) if A233271(n) is even.
In other words, starting from zero, iterate the map x -> {x + 1 + number of nonleading zeros in the binary representation of x}, and note each time whether the result is odd or even: With odd results go one step down, and even results go one step up.
After the zeros at a(0), a(2) and a(4) and -1 at a(1), the terms stay strictly positive for a long time, although from the terms of A257805 it can be seen that the sequence must again fall to the negative side somewhere between n = 541110611 and n = 1051158027 (i.e., A218600(33) .. A218600(34)). Indeed the fourth zero occurs at n = 671605896, and the second negative term right after that as a(671605897) = -1.
The maximum positive value reached prior to the slide into negative territory is 2614822 for a(278998626) and a(278998628). - Hans Havermann, May 23 2015

			We consider 0 to have no nonleading zeros, so first we get to 0 -> 0+1+0 = 1, and 1 is odd, so we go one step down from the starting value a(0)=0, and thus a(1) = -1.
1 has no nonleading zeros, so we get 1 -> 1+1+0 = 2, and 2 is even, so we go one step up, and thus a(2) = 0.
2 has one nonleading zero in binary "10", so we get 2 -> 2+1+1 = 4, and 4 is also even, so we go one step up, and thus a(3) = 1.
4 has two nonleading zeros in binary "100", so we get 4 -> 4+2+1 = 7, 7 is odd, so we go one step down, and thus a(4) = 0.

Antti Karttunen, Table of n, a(n) for n = 0..8727
Hans Havermann, Graph of 2*10^9 terms
Hans Havermann, Detail graph of the eventual crossover to negative terms and a listing of its associated zeros
Wikipedia, Blancmange curve

Cf. A218600, A233271, A233272, A257807, A257808, A257803, A257804, A257805.

Cf. also A218542, A218543, A218789 and A233270 (compare the scatter plots).

A070939 = n->#binary(n)+!n; \\ From M. F. Hasler
A080791 = n->if(n<1,0,(A070939(n)-hammingweight(n)));
A233272 = n->(n + A080791(n) + 1);
A257806_write_bfile(up_to_n) = { my(n,a_n=0,b_n=0); for(n=0, up_to_n, write("b257806.txt", n, " ", a_n); b_n = A233272(b_n); a_n += ((-1)^b_n)); };
A257806_write_bfile(8727);

def A257806_print_upto(n):
  a = 0
  b = 0
  for n in range(n):
     print(b, end=", ")
     ta = a
     c0 = 0
     while ta>0:
         c0 += 1-(ta&1)
         ta >>= 1
     a += 1 + c0
     b += ((2*(1-(a&1))) - 1)
# By Antti Karttunen after Alex Ratushnyak's Python-code for A216431.

(define (A257806 n) (- (A257808 n) (A257807 n)))

;; Using memoizing definec-macro.
(definec (A257806 n) (if (zero? n) n (+  (expt -1 (A233271 n)) (A257806 (- n 1)))))

a(n) = A257808(n) - A257807(n).
a(0) = 0; and for n >= 1, a(n) = a(n-1) + (-1)^A233271(n).
Other identities. For all n >= 0:
a(A218600(n+1)) = -A257805(n).

A257259 a(n) = A218542(n) - A218543(n).

Original entry on oeis.org

1, -1, 0, -1, -1, -3, -1, -6, -8, -10, -7, -4, 2, 2, -14, -90, -318, -896, -2166, -4691, -9298, -17175, -30007, -50261, -81664, -129637, -200973, -301205, -426500, -545183, -561511, -252122, 836543, 3542930, 9312475, 20508985, 40858087, 76119037, 135203839, 232236417, 390709345, 652711050, 1101492173, 1921013671, 3558812217, 7170401227, 15800043812, 37400877416

Offset: 0

Antti Karttunen, May 13 2015

sign

a(n) = the difference between the number of even and odd numbers encountered when traversing from 2^(n+1)-1 to (2^n)-1 by iterating the map A011371: x -> x - (number of 1's in binary representation of x).

Cf. A011371, A218542, A218543, A213709.

Partial sums: A257805.

(define (A257259 n) (- (A218542 n) (A218543 n)))

a(n) = A218542(n) - A218543(n).

cp's OEIS Frontend

A257806 a(n) = A257808(n) - A257807(n).

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A257259 a(n) = A218542(n) - A218543(n).

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