Arie Bos - cp's OEIS frontend

A356426 Even bisection of A003278.

2, 5, 11, 14, 29, 32, 38, 41, 83, 86, 92, 95, 110, 113, 119, 122, 245, 248, 254, 257, 272, 275, 281, 284, 326, 329, 335, 338, 353, 356, 362, 365, 731, 734, 740, 743, 758, 761, 767, 770, 812, 815, 821, 824, 839, 842, 848, 851, 974, 977, 983, 986, 1001, 1004, 1010, 1013, 1055, 1058

Offset: 1

Arie Bos, Aug 07 2022

nonn

Complement sequence of A191107 in A003278.

Cf. A003278, A191107.

def A356426(n): return int(bin((n<<1)-1)[2:],3)+1 # Chai Wah Wu, Sep 07 2022

a(n) = A003278(2n).

A356248 Image of 1 under repeated application of the map k -> (2k-1,2k,2k-1).

Original entry on oeis.org

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

Offset: 0

Arie Bos, Jul 31 2022

nonn

			1 --> 1 2 1 --> 1 2 1 3 4 3 1 2 1 --> 1 2 1 3 4 3 1 2 1 5 6 5 7 8 7 5 6 5 1 2 1 3 4 3 1 2 1 -->...

Cf. A289813.

a(n) = fromdigits(digits(n,3)%2,2) + 1; \\ Kevin Ryde, Jul 31 2022

def aupton(terms):
    a, n = [1], 0
    while len(a) < 3*terms: a, n = a + [(1<Michael S. Branicky, Jul 31 2022

If A(n) = (a(0),a(1),...,a(3^n-1)), then A(n+1) = (A(n),2^n+A(n),A(n)).

a(n) = A289813(n) + 1. - Rémy Sigrist, Jul 31 2022

A356112 Direction of segment n in the E curve of Dekking and McKenna.

Original entry on oeis.org

1, 1, 2, -1, 2, 1, 2, -1, -1, 2, 1, 1, 1, 2, 1, -2, -2, -1, -2, -2, 1, 2, 1, -2, -2, 1, 1, 2, -1, 2, 1, 2, -1, -1, 2, 1, 1, 1, 2, 1, -2, -2, -1, -2, -2, 1, 2, 1, -2, -2, 1, 1, 2, -1, 2, 1, 1, -2, 1, 1, 2, -1, 2, 2, 2, -1, -2, -2, -1, 2, -1, -2, -1, 2, 2, 2, 2, -1, -2, -1, 2, 2, 1, 2, 2, -1, -2, -1, -1, -1, -2, 1, 1, -2, -1, -2, 1, -2, -1, -1, 2, 2, -1, -2

Offset: 0

Arie Bos, Jul 27 2022

sign

On the square grid go one step to the left for a -1, one to the right for a +1, one down for a -2, and one up for a +2. Otherwise stated, replace +-1 with the vector +-(1,0) and +-2 with the vector +-(0,1), then take the running sum to obtain all the vertices of the fractal.

Dekking's "Recurrent sets" published this first, but this "E-curve" was discovered in 1978 by Douglas McKenna.

Douglas M. McKenna, "SquaRecurves, E-Tours, Eddies, and Frenzies: Basic Families of Peano Curves on the Square Grid", in "The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and its History", Mathematical Association of America, 1994, pages 49-73, ISBN 0-88385-516-X.

Arie Bos, Fractal Images as Number Sequences I, arXiv:2207.12942 [cs.CG], 2022. See sec. 3.8, appendix B3.
F. M. Dekking, Recurrent sets, Adv. Math., 44 (1982), 78-104. See example 4.9.

Other curves: A229214, A261180.

If s=[a,b] is a signed permutation, then s(1)=a, s(2)=b, s(-x)=-s(x), a,b,x in {1,2,-1,-2}. Substitution T is defined by T(i) = (i, i, ut, -t, u,   i, ut, -t, -i, ut,   i, i, t, u, t,   -u, -u, -t, -u, -ut,   t, u, i, -ut, -ut), where the signed permutations are defined by i=[1,2], t=[1, -2], u=[2, -1]. The start of the substitution is 1. This means that
T([1,2]x)=([1,2](x),  [1,2](x),  [2,-1][1,-2](x),  -[1,-2](x),  [2,-1](x),     [1,2](x),  [2,-1][1,-2](x),  -[1,-2](x),  -[1,2](x),  [2,-1][1,-2](x),     [1,2](x),  [1,2](x),  [1,-2](x),  [2,-1](x),  [1,-2](x),     -[2,-1](x),  -[2,-1](x),  -[1,-2](x),  -[2,-1](x),  -[2,-1][1,-2](x),     [1,-2](x),  [2,-1](x),  [1,2](x),  -[2,-1][1,-2](x),  -[2,-1][1,-2])(x)),
So T(1)=(1,1,2,-1,2,  1,2,-1,-1,2,  1,1,1,2,1,  -2,-2,-1,-2,-2,  1,2,1,-2,-2) etc.
(See Bos arXiv link, appendix B3.)

A278588 Triangle read by rows: T(p_1,p_2) = maximal period of a decimal fraction (r/s)*(t/u) given that r/s has period p_1 and t/u has period p_2 (1 <= p_1 <= p_2).

Original entry on oeis.org

9, 18, 198, 27, 54, 2997, 36, 396, 108, 39996, 45, 90, 135, 180, 499995, 54, 594, 5994, 3564, 270, 5999994, 63, 126, 189, 252, 315

Offset: 1

N. J. A. Sloane, Dec 03 2016, based on English translations provided by Arie Bos and R. J. Mathar

			Triangle begins (read down columns):
...1.....2....3....4....5.....6......7....(p_2)
1..9....18...27...36...45....54.....63
2......198...54..396...90...594....126
3..........2997..108..135..5994....189
4..............39996..180..3564....252
5..................499995...270....315
...
(p_1)

Arnout Jaspers, Megaperiodieke Breuken Maken, Pythagoras, Number 5, April 2016, pp. 26-29.
Arnout Jaspers, Creating Mega-Periodic Fractions (Loose English translation from _R. J. Mathar_).

Cf. A110807 (diagonal?)

A251714 7-step Fibonacci sequence starting with (0,1,0,0,0,0,0).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 1, 2, 3, 6, 12, 24, 48, 96, 191, 380, 757, 1508, 3004, 5984, 11920, 23744, 47297, 94214, 187671, 373834, 744664, 1483344, 2954768, 5885792, 11724287, 23354360, 46521049, 92668264, 184591864, 367700384, 732446000, 1459006208, 2906288129

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251710, A251711, A251712, A251713.

NB. see A251713 for the program and apply it to 0 1 0 0 0 0 0.

LinearRecurrence[Table[1, {7}], {0, 1, 0, 0, 0, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).

G.f.: x*(-1+x+x^2+x^3+x^4+x^5)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7) . - R. J. Mathar, Mar 28 2025

A251713 7-step Fibonacci sequence starting with (0,0,1,0,0,0,0).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 1, 2, 4, 7, 14, 28, 56, 112, 223, 444, 884, 1761, 3508, 6988, 13920, 27728, 55233, 110022, 219160, 436559, 869610, 1732232, 3450544, 6873360, 13691487, 27272952, 54326744, 108216929, 215564248, 429396264, 855341984, 1703810608, 3393929729

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251710, A251711, A251712, A251714.

(see www.jsoftware.com) First construct the generating matrix
   [M=: (#.@}: + {:)\"1&.|: <:/~i.7
 1  1  1  1  1  1  1
 1  2  2  2  2  2  2
 2  3  4  4  4  4  4
 4  6  7  8  8  8  8
 8 12 14 15 16 16 16
16 24 28 30 31 32 32
32 48 56 60 62 63 64
Given that matrix, one can produce the first 7*150 numbers by
, M(+/ . *)^:(i.150) 0 0 1 0 0 0 0x

LinearRecurrence[Table[1, {7}], {0, 0, 1, 0, 0, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).

G.f.: x^2*(-1+x+x^2+x^3+x^4)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7) . - R. J. Mathar, Mar 28 2025

A251712 7-step Fibonacci sequence starting with (0,0,0,1,0,0,0).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 4, 8, 15, 30, 60, 120, 239, 476, 948, 1888, 3761, 7492, 14924, 29728, 59217, 117958, 234968, 468048, 932335, 1857178, 3699432, 7369136, 14679055, 29240152, 58245336, 116022624, 231112913, 460368648, 917037864, 1826706592, 3638734129

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251710, A251711, A251713, A251714.

NB. see A251713 for the program and apply it to 0 0 0 1 0 0 0.

LinearRecurrence[Table[1, {7}], {0, 0, 0, 1, 0, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).

G.f.: x^3*(-1+x+x^2+x^3)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7) . - R. J. Mathar, Mar 28 2025

A251711 7-step Fibonacci sequence starting with (0,0,0,0,1,0,0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 2, 4, 8, 16, 31, 62, 124, 247, 492, 980, 1952, 3888, 7745, 15428, 30732, 61217, 121942, 242904, 483856, 963824, 1919903, 3824378, 7618024, 15174831, 30227720, 60212536, 119941216, 238918608, 475917313, 948010248, 1888402472, 3761630113

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251710, A251712, A251713, A251714.

NB. see A251713 for the program and apply it to 0 0 0 0 1 0 0.

LinearRecurrence[Table[1, {7}], {0, 0, 0, 0, 1, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).
G.f.: x^4*(-1+x+x^2)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7) . - R. J. Mathar, Mar 28 2025
a(n) = A066178(n-4)-A066178(n-5)-A066178(n-6). - R. J. Mathar, Mar 28 2025

A251710 7-step Fibonacci sequence starting with (0,0,0,0,0,1,0).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 2, 4, 8, 16, 32, 63, 126, 251, 500, 996, 1984, 3952, 7872, 15681, 31236, 62221, 123942, 246888, 491792, 979632, 1951392, 3887103, 7742970, 15423719, 30723496, 61200104, 121908416, 242837200, 483723008, 963558913, 1919374856, 3823325993

Offset: 0

Arie Bos, Dec 07 2014

a(n+7) equals the number of n-length binary words avoiding runs of zeros of lengths 7i+6, (i=0,1,2,...). - Milan Janjic, Feb 26 2015

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Other 7-step Fibonacci sequences are A066178, A104621, A122189, A251711, A251712, A251713, A251714.

NB. see A251713 for the program and apply it to 0 0 0 0 0 1 0.

LinearRecurrence[Table[1, {7}], {0, 0, 0, 0, 0, 1, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+7) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5) + a(n+6).
G.f.: x^5*(x-1)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7) . - R. J. Mathar, Mar 28 2025
a(n) = A066178(n-5)-A066178(n-6). - R. J. Mathar, Mar 28 2025

A251654 4-step Fibonacci sequence starting with 0, 1, 1, 0.

Original entry on oeis.org

0, 1, 1, 0, 2, 4, 7, 13, 26, 50, 96, 185, 357, 688, 1326, 2556, 4927, 9497, 18306, 35286, 68016, 131105, 252713, 487120, 938954, 1809892, 3488679, 6724645, 12962170, 24985386, 48160880, 92833081, 178941517, 344920864, 664856342, 1281551804

Offset: 0

Arie Bos, Dec 06 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251655, A251656, A251672, A251703, A251704, A251705.

NB. see A251655 for the program and apply it to 0,1,1,0.

LinearRecurrence[Table[1, {4}], {0, 1, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: x*(-1+2*x^2)/(-1+x+x^2+x^3+x^4). - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+2)-2*A000078(n). - R. J. Mathar, Mar 28 2025

cp's OEIS Frontend

User: Arie Bos

A356426 Even bisection of A003278.

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A356248 Image of 1 under repeated application of the map k -> (2k-1,2k,2k-1).

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PARI

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A356112 Direction of segment n in the E curve of Dekking and McKenna.

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A278588 Triangle read by rows: T(p_1,p_2) = maximal period of a decimal fraction (r/s)*(t/u) given that r/s has period p_1 and t/u has period p_2 (1 <= p_1 <= p_2).

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A251714 7-step Fibonacci sequence starting with (0,1,0,0,0,0,0).

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J

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A251713 7-step Fibonacci sequence starting with (0,0,1,0,0,0,0).

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J

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A251712 7-step Fibonacci sequence starting with (0,0,0,1,0,0,0).

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J

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A251711 7-step Fibonacci sequence starting with (0,0,0,0,1,0,0).

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J

Mathematica

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A251710 7-step Fibonacci sequence starting with (0,0,0,0,0,1,0).

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