%I #26 Apr 06 2024 07:57:53
%S 1,3,4,6,8,11,13,14,16,19,20,22,25,27,29,30,32,35,36,38,40,43,45,46,
%T 49,51,52,54,57,59,61,62,64,67,68,70,72,75,77,78,80,83,84,86,89,91,93,
%U 94,97,99,100,102,104,107,109,110,113,115,116,118,121,123,125
%N Starting positions of runs in the paperfolding sequence A014707.
%C A "run" is a maximal block of consecutive identical terms. The paperfolding sequence A014707 is more usually indexed starting at position 1, not 0, and this choice is reflected in the sequence (cf. A034947).
%H M. Bunder, B. Bates, and S. Arnold, <a href="https://doi.org/10.1017/S0004972724000169">The summed paperfolding sequence</a>, Bull. Aust. Math. Soc. (2024).
%H Kevin Ryde, <a href="http://user42.tuxfamily.org/dragon/index.html">Iterations of the Dragon Curve</a>, see index "TurnRunStart" with a(n) = TurnRunStart(n-1).
%H Jeffrey Shallit, <a href="/A371594/a371594.pdf">Automaton for A371594</a>.
%F The automaton accompanying this entry accepts exactly the base-2 representations of the terms of this sequence.
%F a(n) = 2*n-1 - ((n + A014707(n-2)) mod 2), for n >= 2. - _Kevin Ryde_, Mar 28 2024
%e The first few terms of A014707 are 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, and runs begin at positions 1, 3, 4, 6, 8, 11, 13, 14, ...
%t Abs@ SplitBy[Array[# KroneckerSymbol[-1, #] &, 120], Sign][[All, 1]] (* _Michael De Vlieger_, Mar 28 2024 *)
%o (Python) # DFA transition function and simulation
%o d = { (0,0):0, (0,1):1, (1,0):2, (1,1):3, (2,0):4, (2,1):5,
%o (3,0):6, (3,1):7, (4,0):4, (4,1):5, (5,0):2, (5,1):3,
%o (6,0):0, (6,1):1, (7,0):6, (7,1):7 }
%o def ok(n):
%o q, w = 0, map(int, bin(n)[2:])
%o for c in w: q = d[q, c]
%o return q in {1, 3, 4, 6}
%o print([k for k in range(126) if ok(k)]) # _Michael S. Branicky_, Mar 28 2024
%o (Python) # using formula and function in A014707
%o def a(n): return 2*n-1 - (n + A014707(n-2))%2 if n>=2 else 1
%o print([a(n) for n in range(1, 64)]) # _Michael S. Branicky_, Mar 29 2024
%o (PARI) a(n) = if(n==1,1, n--; 2*n + bitxor(bittest(n,0), bittest(n,valuation(n,2)+1))); \\ _Kevin Ryde_, Apr 06 2024
%Y Cf. A014707, A034947.
%K nonn,easy
%O 1,2
%A _Jeffrey Shallit_, Mar 28 2024