cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A357065 Numbers k with the following property: the value A091839(k+1) is not a 1 that is obtained from smoothing A091579.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 29, 31, 33, 34, 35, 37, 39, 40, 41, 42, 43, 45, 47, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 63, 65, 66, 67, 69, 71, 73, 74, 75, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 91, 93, 95, 97, 98, 99, 101, 103
Offset: 1

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Author

Levi van de Pol, Sep 10 2022

Keywords

Comments

This sequence is the function iota1 in the article "The first occurrence of a number in Gijswijt's sequence" (page 21). For the connection with smoothing, see Subsection 8.1.

Examples

			14 is not a term since A091839(14+1) is a 1 obtained from smoothing: in A091579, the eleventh value is 4, which is replaced by 3,1 to obtain the fourteenth and fifteenth terms of A091839.
		

Crossrefs

A091579 Lengths of suffix blocks associated with A090822.

Original entry on oeis.org

1, 3, 1, 9, 4, 24, 1, 3, 1, 9, 4, 67, 1, 3, 1, 9, 4, 24, 1, 3, 1, 9, 4, 196, 3, 1, 9, 4, 24, 1, 3, 1, 9, 4, 68, 3, 1, 9, 4, 24, 1, 3, 1, 9, 4, 581, 3, 1, 9, 4, 25, 3, 1, 9, 4, 67, 1, 3, 1, 9, 4, 24, 1, 3, 1, 9, 4, 196, 3, 1, 9, 4, 24, 1, 3, 1, 9, 4, 68, 3, 1, 9, 4, 24, 1, 3, 1, 9, 4, 1731, 3, 1, 9, 4, 24
Offset: 1

Views

Author

N. J. A. Sloane, Mar 05 2004

Keywords

Comments

The suffix blocks are what is called "glue string" in the paper by Gijswijt et al (2007). Roughly speaking, these are the terms >= 2 appended before the sequence (A090822) goes on with a(n+1) = 1 followed by all other initial terms a(2..n), cf. Example. The concatenation of these glue strings yields A091787. - M. F. Hasler, Aug 08 2018

Examples

			From _M. F. Hasler_, Aug 09 2018: (Start)
In sequence A090822, after the initial (1, 1) follows the first suffix block or glue string (2) of length a(1) = 1. This is followed by A090822(4) = 1 which indicates that the suffix block has ended, and the whole sequence A090822(1..3) up to and including this suffix block is repeated: A090822(4..6) = A090822(1..3).
Then A090822 goes on with (2, 2, 3, 1, ...), which tells that the second suffix block is A090822(7..9) = (2, 2, 3) of length a(2) = 3, whereafter the sequence starts over again: A090822(10..18) = A090822(1..9). (End)
		

Crossrefs

Cf. A090822, A091587 (records). For a smoothed version see A091839.
Cf. A091787 for the concatenation of the glue strings.

Programs

  • Python
    # compute curling number of L
    def curl(L):
        n = len(L)
        m = 1 #max nr. of repetitions at the end
        k = 1 #length of repeating block
        while(k*(m+1) <= n):
            good = True
            i = 1
            while(i <= k and good):
                for t in range(1, m+1):
                    if L[-i-t*k] != L[-i]:
                        good = False
                i = i+1
            if good:
                m = m+1
            else:
                k = k+1
        return m
    # compute lengths of first n glue strings
    def A091579_list(n):
        Promote = [1] #Keep track of promoted elements
        L = [2]
        while len(Promote) <= n:
            c = curl(L)
            if c < 2:
                Promote = Promote+[len(L)+1]
                c = 2
            L = L+[c]
        return [Promote[i+1]-Promote[i] for i in range(n)]
    # Dion Gijswijt, Oct 08 2015

A091588 A smoothed version of A091587.

Original entry on oeis.org

1, 3, 8, 24, 67, 195, 580, 1730, 5179, 15533, 46578, 139712, 419115, 1257319, 3771930, 11315764, 33947261, 101841751, 305525228, 916575642, 2749726883
Offset: 0

Views

Author

N. J. A. Sloane, Mar 05 2004

Keywords

Comments

Each term is roughly 3 times the previous term.

Crossrefs

Records in A091839. Cf. A090822, A091587, A091587, A217590.

Formula

a(n+1) = a(n) + A357063(n+1) + A091840(n+1). See Conjecture 4.2 of F. J. van de Bult et al., proved p. 54 of Levi van de Pol. - Levi van de Pol, Nov 04 2022

Extensions

a(10)-a(13) from Allan Wilks, Mar 10 2004
a(14)-a(20) from Alexander Staunton, Apr 09 2022
Removed an incorrect program. - N. J. A. Sloane, Aug 20 2022

A095828 Smoothed lengths of the B blocks in analysis of A090822.

Original entry on oeis.org

1, 3, 9, 19, 46, 93, 189, 379, 782, 1565, 3133, 6267, 12542, 25085, 50173, 100347, 200761, 401523, 803049, 1606099, 3212206, 6424413, 12848829, 25697659, 51395342, 102790685, 205581373, 411162747, 822325502, 1644651005, 3289302013, 6578604027, 13157208249
Offset: 1

Views

Author

N. J. A. Sloane, Jul 10 2004

Keywords

Crossrefs

Formula

a(1) = 1; for n > 1, a(n+1) = 2*a(n) + A091839(n).
This roughly doubles at each step and a(n) -> 1.5317006328915480... * 2^n.
Showing 1-4 of 4 results.