Robert H Barbour - cp's OEIS frontend

A126979 a(n) = 24*n + 233.

233, 257, 281, 305, 329, 353, 377, 401, 425, 449, 473, 497, 521, 545, 569, 593, 617, 641, 665, 689, 713, 737, 761, 785, 809, 833, 857, 881, 905, 929, 953, 977, 1001, 1025, 1049, 1073, 1097, 1121, 1145, 1169, 1193, 1217, 1241, 1265, 1289, 1313, 1337, 1361

Offset: 0

Robert H Barbour, Mar 20 2007, Jun 12 2007

Superhighway created by 'LQTL Ant' L45R135L45R135 from iteration 233 where the Ant moves in a 'Moore neighborhood' (nine cells), the L indicates a left turn, the R a right turn, and the numerical value is the turn angle in degrees.

P. Sakar, "A Brief History of Cellular Automata," ACM Computing Surveys, vol. 32, 2000.
S. Wolfram, A New Kind of Science, 1st ed. Il.: Wolfram Media Inc., 2002.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A031041, A017581, A126978, A126980. Has many terms in common with A031041.

a:=[233, 257];; for n in [3..60] do a[n]:=2*a[n-1]-a[n-2]; od; a; # G. C. Greubel, May 28 2019

I:=[233, 257]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..60]]; // G. C. Greubel, May 28 2019

Table[24*n + 233, {n, 0, 60}] (* Stefan Steinerberger, Jun 17 2007 *)
LinearRecurrence[{2,-1}, {233,257}, 60] (* G. C. Greubel, May 28 2019 *)

my(x='x+O('x^60)); Vec((233-209*x)/(1-x)^2) \\ G. C. Greubel, May 28 2019

((233-209*x)/(1-x)^2).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, May 28 2019

From Chai Wah Wu, May 30 2016: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: (233 - 209*x)/(1 - x)^2. (End)
E.g.f.: (233 + 24*x)*exp(x). - G. C. Greubel, May 28 2019

More terms from Stefan Steinerberger, Jun 17 2007

A126980 a(n) = 14*n + 47.

Original entry on oeis.org

47, 61, 75, 89, 103, 117, 131, 145, 159, 173, 187, 201, 215, 229, 243, 257, 271, 285, 299, 313, 327, 341, 355, 369, 383, 397, 411, 425, 439, 453, 467, 481, 495, 509, 523, 537, 551, 565, 579, 593, 607, 621, 635, 649, 663, 677, 691, 705, 719, 733, 747

Offset: 0

Robert H Barbour, Mar 20 2007, Jun 12 2007

Superhighway created by 'LQTL Ant' L90R135L90R135 from iteration 47 where the Ant moves in a 'Moore neighborhood' (nine cells), the L indicates a left turn, the R a right turn, and the numerical value is the turn angle (in degrees) at each iteration.

P. Sakar, "A Brief History of Cellular Automata," ACM Computing Surveys, vol. 32, 2000.
S. Wolfram, A New Kind of Science, 1st ed. Il.: Wolfram Media Inc., 2002.

Diana Mecum, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017041, A019430, A094867, A126978, A126979.

[14*n + 47: n in [0..60] ]; // Vincenzo Librandi, Jul 18 2011

Table[14*n + 47, {n, 0, 60}] (* Stefan Steinerberger, Jun 17 2007 *)
LinearRecurrence[{2,-1}, {47,61}, 50] (* G. C. Greubel, May 30 2016 *)

From Chai Wah Wu, May 30 2016: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: (-33*x + 47)/(x - 1)^2. (End)
E.g.f.: (47 + 14*x)*exp(x). - G. C. Greubel, May 30 2016

More terms from Stefan Steinerberger and Diana L. Mecum, Jun 17 2007

A127547 a(n) = 13*n + 4.

Original entry on oeis.org

4, 17, 30, 43, 56, 69, 82, 95, 108, 121, 134, 147, 160, 173, 186, 199, 212, 225, 238, 251, 264, 277, 290, 303, 316, 329, 342, 355, 368, 381, 394, 407, 420, 433, 446, 459, 472, 485, 498, 511, 524, 537, 550, 563, 576, 589, 602, 615, 628, 641, 654, 667, 680, 693, 706, 719

Offset: 0

Robert H Barbour, Apr 01 2007

Superhighway created by 'LQTL Ant' L90R90L45R45 from iteration 4 where the Ant moves in a 'Moore neighborhood' (nine cells), the L indicates a left turn, the R a right turn, and the numerical value is the size of the turn (in degrees) at each iteration.

Ant Farm algorithm available from Robert H Barbour.

P. Sakar, "A Brief History of Cellular Automata," ACM Computing Surveys, vol. 32, pp. 80-107, 2000.

G. C. Greubel, Table of n, a(n) for n = 0..5000
C. Langton, Studying Artificial Life with Cellular Automata, Physica D: Nonlinear Phenomena, vol. 22, pp. 120-149, 1986.
James Propp, Further Ant-ics, Mathematical Intelligencer, 16 pp. 37-42, 1994.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

A subsequence of A092464.
Cf. A126979, A126980.
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), this sequence (q=4), A154609 (q=5), A186113 (q=6), A269044 (q=7), A269100 (q=11).

[13*n+4: n in [0..60]]; // G. C. Greubel, May 31 2024

Range[4,1000,13] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

[13*n+4 for n in range(61)] # G. C. Greubel, May 31 2024

From Elmo R. Oliveira, Mar 21 2024: (Start)
G.f.: (4+9*x)/(1-x)^2.
E.g.f.: (4 + 13*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

Edited by N. J. A. Sloane, May 10 2007

A126978 a(n) = 104*n + 9977.

Original entry on oeis.org

9977, 10081, 10185, 10289, 10393, 10497, 10601, 10705, 10809, 10913, 11017, 11121, 11225, 11329, 11433, 11537, 11641, 11745, 11849, 11953, 12057, 12161, 12265, 12369, 12473, 12577, 12681, 12785, 12889, 12993, 13097, 13201, 13305, 13409, 13513, 13617, 13721, 13825

Offset: 0

Robert H Barbour, Mar 20 2007, Jun 12 2007

Langton's Ant Superhighway, the start point (9977th iteration, J. Propp) and the period length for the Superhighway (104).

B. D. Swan, Table of n, a(n) for n = 0..10000
C. Langton, Studying Artificial Life with Cellular Automata, Physica D: Nonlinear Phenomena, Vol. 22, 1986, pp. 120-149.
Ed Pegg Jr, 2D Turing Machines, 2004.
James Propp, Further Ant-ics, Mathematical Intelligencer, Vol. 16, 1994, pp. 37-42.
P. Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys. Vol. 32, No. 1, Mar 01 2000, pp. 80-107.
S. Wolfram, 2D Turing Machines.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A102127, A102358, A102369, A126979, A126980.

[104*n + 9977: n in [0..40]]; // Vincenzo Librandi, Sep 10 2015

104*Range[0,40]+9977 (* or *) LinearRecurrence[{2,-1},{9977,10081},40] (* Harvey P. Dale, Dec 16 2011 *)
CoefficientList[Series[(9977 - 9873 x)/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 10 2015 *)

a(n)=104*n+9977 \\ Charles R Greathouse IV, Oct 07 2015

a(0)=9977, a(1)=10081, a(n) = 2*a(n-1) - a(n-2). - Harvey P. Dale, Dec 16 2011
G.f.: (9977 - 9873*x)/(1-x)^2. - Vincenzo Librandi, Sep 10 2015
E.g.f.: exp(x)*(9977 + 104*x). - Elmo R. Oliveira, Dec 08 2024

A106624 Expansion of g.f.: (1 - x^2 + x^3)/((1-x^2)(1-2x^2)).

Original entry on oeis.org

1, 0, 2, 1, 4, 3, 8, 7, 16, 15, 32, 31, 64, 63, 128, 127, 256, 255, 512, 511, 1024, 1023, 2048, 2047, 4096, 4095, 8192, 8191, 16384, 16383, 32768, 32767, 65536, 65535, 131072, 131071, 262144, 262143, 524288, 524287, 1048576, 1048575, 2097152

Offset: 0

Robert H Barbour, May 10 2005

Cumulative column frequency of occurrence of 0's and 1's iterated in a binary tree where each node in the tree holds a value of 0 or 1, beginning with a count of 1.

Douglas Comer, Ubiquitous B-Tree, ACM Computing Surveys (CSUR), (1979), Volume 11 Issue 2.
Huffman, D. A., A method for the construction of minimum redundancy codes, Proc. IRE 40 (1951), 1098-1101.
Knuth, D. E., Dynamic Huffman coding. J. Algorithms 6 (1985), 163-180.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
G. C. Barnes et al., Balanced sample Binary Trees, ACM SIGCSE Bulletin, Volume 37 Issue 1, 2005 pp. 166-170.
P. N. Fenwick, Cumulative Frequency, Software: Practice and Experience, 1994.
C. Witteveen, Balanced Binary Trees, Slides.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A016116, A014535, A037026, A058518 - A058521, A000079 (bisection), A000225 (bisection).

[2^Floor(n/2) + Floor((-1)^n - 1)/2: n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

A106624 := proc(N)
    2^floor(n/2)+((-1)^n-1)/2 ;
end proc:
seq(A106624(n),n=0..20) ; # R. J. Mathar, Apr 14 2018

Table[2^Floor[n/2] +Floor[(-1)^n-1]/2, {n,0,50}] (* G. C. Greubel, Feb 19 2019 *)

vector(50,n, n--; 2^floor(n/2) +floor((-1)^n-1)/2) \\ G. C. Greubel, Feb 19 2019

[2^floor(n/2) +floor((-1)^n-1)/2 for n in (0..50)] # G. C. Greubel, Feb 19 2019

a(n) = 2^floor(n/2) + floor((-1)^n - 1)/2. - N. J. A. Sloane, May 15 2005

New definition from N. J. A. Sloane, May 15 2008

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A108717 An erroneous sequence.

Original entry on oeis.org

11111111, 11111112, 11111212, 11111222, 11111212, 11121222, 11122222, 11122221

Offset: 0

Robert H Barbour, Jun 20 2005

dead

Old name was: A model of LQTL "possible world lines" as a balanced binary tree with four successors (S4S). Each path through the tree creates a unique "word" describing the linked nodes. Four iterations generate the sequence which is extensible. Qubits mapped to a balanced binary tree.

For this sequence qubits have been represented as 11, 12, 21, 22, rather than the more common 00, 01, 11, 10. With the latter encoding the sequence would be 00000000, 00000001, 00000101, 00000111, 00000101, 00010111, 00011111, 00011110.

Robert Barbour, 2D Four-Color Cellular Automaton, NKS 2006 Wolfram Science Conference. See presentation materials, slide 14. Note that a(4) in this sequence entry has a typo: according to the slides, it should be 11121212, representing the bit sequence 00010101.

A102369 Intervals between successive arrivals of Langton's Ant at the origin.

Original entry on oeis.org

4, 4, 8, 36, 8, 36, 44, 44, 92, 92, 8, 8, 8, 8, 36, 8, 36, 184, 4, 2752, 732, 8, 2016, 596, 1284, 4, 4, 4, 8

Offset: 0

Robert H Barbour, Feb 22 2005, corrected Feb 27 2005

Ant Farm algorithm available from Robert H Barbour.

First differences of A102358. - Felix Fröhlich, Jul 26 2016

Christopher G. Langton et al., (1990) Artificial Life II. Addison-Wesley, Reading, MA., USA

P. Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys. Vol. 32 No. 1 March (2000).

Cf. A102127, A102358.

A102358 Finite sequence of iterations at which Langton's Ant passes through the origin.

Original entry on oeis.org

0, 4, 8, 16, 52, 60, 96, 140, 184, 276, 368, 376, 384, 392, 428, 436, 472, 656, 660, 3412, 4144, 4152, 6168, 6764, 8048, 8052, 8056, 8060, 8068

Offset: 1

Robert H Barbour, Feb 21 2005, corrected Feb 27 2005

Ant Farm algorithm available from bbarbour(AT)unitec.ac.nz.

Numbers n such that A274369(n) = A274370(n) = 0. - Felix Fröhlich, Jul 26 2016

Christopher G. Langton et al., (1990) Artificial Life II. Addison-Wesley, Reading, MA., USA

P. Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys. Vol. 32 No. 1 March (2000).

Cf. A102127, A102369, A274369, A274370.

A102110 Iterations during which LQTL cellular automaton passes through the origin.

Original entry on oeis.org

4, 8, 12, 28, 32, 36, 40, 56, 60, 64, 68, 72, 76, 116, 120, 136, 140, 144, 148, 152, 164, 168, 172, 180, 184, 224, 228, 232, 236, 240, 244, 332, 336, 376, 380, 384, 388, 392, 396, 412, 416, 420, 424, 440, 444, 448, 452, 456, 460, 500, 504, 592, 596, 600, 604

Offset: 1

Robert H Barbour, Feb 14 2005

Robert H. Barbour, Two-dimensional Four Color Cellular Automaton: Surface Explorations, Complex Systems, 17 (2007), 103-112.
Palash Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys, Vol. 32 (2000), No. 1 March, p. 80-107.

Cf. A094266, A094867, A099423.

Cumulative sums of A102127.

Terms a(19) and beyond from Andrey Zabolotskiy, Jan 06 2023

A099423 Lean quaternary temporal logic [LQTL] cumulative column frequencies of LQTL logic in A094266.

Original entry on oeis.org

1, 0, 0, 0, 2, 1, 0, 0, 3, 3, 1, 0, 4, 6, 4, 1, 6, 10, 10, 5, 12, 16, 20, 15, 28, 28, 36, 35, 64, 56, 64, 71, 136, 120, 120, 135, 272, 256, 240, 255, 528, 528, 496, 495, 1024, 1056, 1024, 991, 2016, 2080, 2080, 2015, 4032, 4096, 4160, 4095, 8128, 8128, 8256, 8255

Offset: 0

Robert H Barbour, Nov 17 2004; corrected Jun 10 2005

Appears to satisfy the recurrence: a(n) = 5*a(n-4) - 10*a(n-8) + 10*a(n-12) - 4*a(n-16). - Chai Wah Wu, May 25 2016

This is a table read by rows of length 4. Every row is formed from the corresponding row of the table A094266 according to the rule: a, b, c, d -> b, c, d, a-1. - Andrey Zabolotskiy, Jan 06 2023

Robert H. Barbour, Two-dimensional Four Color Cellular Automaton: Surface Explorations, Complex Systems, 17 (2007), 103-112.

Cf. A094266 (first differences row-wise).

cp's OEIS Frontend

User: Robert H Barbour

A126979 a(n) = 24*n + 233.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A126980 a(n) = 14*n + 47.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A127547 a(n) = 13*n + 4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A126978 a(n) = 104*n + 9977.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A106624 Expansion of g.f.: (1 - x^2 + x^3)/((1-x^2)*(1-2*x^2)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A108717 An erroneous sequence.

Views

Author

Keywords

Comments

Links

A102369 Intervals between successive arrivals of Langton's Ant at the origin.

A106624 Expansion of g.f.: (1 - x^2 + x^3)/((1-x^2)(1-2x^2)).