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A274575 For m=1,2,3,... write all the 2^m binary vectors of length m in increasing order, and replace each vector with (number of 1's) - (number of 0's). Start with an initial 0 for the empty vector.

%I A274575 #44 Apr 21 2024 13:24:08
%S A274575 0,-1,1,-2,0,0,2,-3,-1,-1,1,-1,1,1,3,-4,-2,-2,0,-2,0,0,2,-2,0,0,2,0,2,
%T A274575 2,4,-5,-3,-3,-1,-3,-1,-1,1,-3,-1,-1,1,-1,1,1,3,-3,-1,-1,1,-1,1,1,3,
%U A274575 -1,1,1,3,1,3,3,5,-6,-4,-4,-2,-4,-2,-2,0,-4,-2,-2,0,-2,0,0,2,-4,-2,-2,0,-2,0,0,2,-2,0,0,2,0,2,2,4,-4,-2,-2,0,-2,0,0,2,-2,0,0,2,0,2,2,4,-2,0,0,2,0,2,2,4,0
%N A274575 For m=1,2,3,... write all the 2^m binary vectors of length m in increasing order, and replace each vector with (number of 1's) - (number of 0's). Start with an initial 0 for the empty vector.
%C A274575 This is the sequence of To-And-Fro positions: Positions of all backward-forward combinations in lexicographical order when assigning -1 to a backward move and +1 to a forward move and starting at 0.
%C A274575 -a(n) are the slopes of the different segments, from left to right, of the successive steps in the construction of the Takagi (a.k.a. Blancmange) function. - _Javier Múgica_, Dec 31 2017
%H A274575 John Tyler Rascoe, <a href="/A274575/b274575.txt">Table of n, a(n) for n = 0..8190</a>
%F A274575 a(2*n + 1) = a(n) - 1; a(2*n + 2) = a(n) + 1.
%e A274575 Terms a(3) to a(6) correspond to the binary vectors 00, 01, 10, 11, which get replaced by -2, 0, 0, 2, respectively. Terms a(7) to a(14) correspond to the binary vectors 000, 001, ..., 111 which get replaced by -3, -1, ..., 3. a(0) = 0
%e A274575 a(1) = a('backward') = -1
%e A274575 a(2) = a('forward') = +1
%e A274575 a(3) = a('backward and backward') = -2
%e A274575 a(4) = a('backward and forward') = 0
%e A274575 a(5) = a('forward and backward') = 0
%e A274575 a(6) = a('forward and forward') = +2
%e A274575 a(7) = a('backward, backward and backward') = -3
%e A274575 a(8) = a('backward, backward and forward') = -1
%e A274575 Arranged as a tree read by rows:
%e A274575                ______0______
%e A274575               /             \
%e A274575           __-1__           __1__
%e A274575          /      \         /     \
%e A274575        -2        0       0       2
%e A274575        / \      / \     / \     / \
%e A274575      -3  -1   -1   1  -1   1   1   3
%e A274575 . - _John Tyler Rascoe_, Sep 23 2023
%o A274575 (BASIC)
%o A274575 Dim a(2*k+2)
%o A274575 a(0) = 0
%o A274575 For n = 0 To k
%o A274575   a(2 * n + 1) = a(n) - 1
%o A274575   a(2 * n + 2) = a(n) + 1
%o A274575 Next n
%o A274575 (Python)
%o A274575 def A274575_list(nmax):
%o A274575     A = [0]
%o A274575     for n in range(0,nmax):
%o A274575         A.append(A[n//2]-(-1)**n)
%o A274575     return(A)
%o A274575 print(A274575_list(119)) # _John Tyler Rascoe_, Sep 23 2023
%Y A274575 Cf. A037861.
%K A274575 sign,easy
%O A274575 0,4
%A A274575 _Hans G. Oberlack_, Jun 28 2016
%E A274575 Edited by _N. J. A. Sloane_, Jul 27 2016