A371594 Starting positions of runs in the paperfolding sequence A014707.

1, 3, 4, 6, 8, 11, 13, 14, 16, 19, 20, 22, 25, 27, 29, 30, 32, 35, 36, 38, 40, 43, 45, 46, 49, 51, 52, 54, 57, 59, 61, 62, 64, 67, 68, 70, 72, 75, 77, 78, 80, 83, 84, 86, 89, 91, 93, 94, 97, 99, 100, 102, 104, 107, 109, 110, 113, 115, 116, 118, 121, 123, 125

Offset: 1

Author

Jeffrey Shallit, Mar 28 2024

Keywords

• nonn
• easy

Comments

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).

Examples

			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, ...
		

Links

• M. Bunder, B. Bates, and S. Arnold, The summed paperfolding sequence, Bull. Aust. Math. Soc. 110 (2024), 189-198.
• Kevin Ryde, Iterations of the Dragon Curve, see index "TurnRunStart" with a(n) = TurnRunStart(n-1).
• Jeffrey Shallit, Automaton for A371594.

Crossrefs

Cf. A014707, A034947.

Programs

Mathematica

Abs@ SplitBy[Array[#  KroneckerSymbol[-1, #] &, 120], Sign][[All, 1]] (* Michael De Vlieger, Mar 28 2024 *)

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

Python

# DFA transition function and simulation
d = { (0,0):0, (0,1):1, (1,0):2, (1,1):3, (2,0):4, (2,1):5,
      (3,0):6, (3,1):7, (4,0):4, (4,1):5, (5,0):2, (5,1):3,
      (6,0):0, (6,1):1, (7,0):6, (7,1):7 }
def ok(n):
    q, w = 0, map(int, bin(n)[2:])
    for c in w: q = d[q, c]
    return q in {1, 3, 4, 6}
print([k for k in range(126) if ok(k)]) # Michael S. Branicky, Mar 28 2024

Python

# using formula and function in A014707
def a(n): return 2*n-1 - (n + A014707(n-2))%2 if n>=2 else 1
print([a(n) for n in range(1, 64)]) # Michael S. Branicky, Mar 29 2024

Formula

The automaton accompanying this entry accepts exactly the base-2 representations of the terms of this sequence.

a(n) = 2*n-1 - ((n + A014707(n-2)) mod 2), for n >= 2. - Kevin Ryde, Mar 28 2024