A126980 -id:A126980

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A127547

a(n) = 13*n + 4.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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.

REFERENCES

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

LINKS

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).

FORMULA

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)

MATHEMATICA

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

PROG

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

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

CROSSREFS

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).

KEYWORD

easy,nonn

AUTHOR

Robert H Barbour, Apr 01 2007

EXTENSIONS

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

STATUS

approved

A126978

a(n) = 104*n + 9977.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

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, pp. 120-149, 1986.

Ed Pegg Jr, 2D Turing Machines .

James Propp, Further Ant-ics, Mathematical Intelligencer, 16 pp. 37-42, 1994.

P. Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys. Vol. 32 No. Mar 01 2000.

S. Wolfram, 2D Turing Machines.

Index entries for linear recurrences with constant coefficients, signature (2, -1).

FORMULA

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

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Robert H Barbour, Mar 20 2007, Jun 12 2007

STATUS

approved

A126979

a(n) = 24*n + 233.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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.

REFERENCES

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.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

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

MATHEMATICA

Table[24*n + 233, {n, 0, 60}] (* Stefan Steinerberger, Jun 17 2007 *)

LinearRecurrence[{2, -1}, {233, 257}, 60] (* G. C. Greubel, May 28 2019 *)

PROG

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

(Magma) 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

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

(GAP) 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

CROSSREFS

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

KEYWORD

easy,nonn

AUTHOR

Robert H Barbour, Mar 20 2007, Jun 12 2007

EXTENSIONS

More terms from Stefan Steinerberger, Jun 17 2007

STATUS

approved

A163676

Triangle T(n,m) = 4mn + 2m + 2n - 1 read by rows.

+10
2

7, 13, 23, 19, 33, 47, 25, 43, 61, 79, 31, 53, 75, 97, 119, 37, 63, 89, 115, 141, 167, 43, 73, 103, 133, 163, 193, 223, 49, 83, 117, 151, 185, 219, 253, 287, 55, 93, 131, 169, 207, 245, 283, 321, 359, 61, 103, 145, 187, 229, 271, 313, 355, 397, 439, 67, 113, 159

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

2 + T(n,m) = (2*n+1)*(2*m+1) are composite numbers. - clarified by R. J. Mathar, Oct 16 2009

First column: A016921, second column: A017305, third column: A126980. - Vincenzo Librandi, Nov 21 2012

LINKS

Vincenzo Librandi, Rows n = 1..100, flattened

FORMULA

T(n,m) = A155151(n,m) - 3 = A155156(n,m) - 1. - R. J. Mathar, Oct 16 2009

EXAMPLE

Triangle begins:

13, 23;

19, 33, 47;

25, 43, 61, 79;

31, 53, 75, 97, 119;

37, 63, 89, 115, 141, 167;

43, 73, 103, 133, 163, 193, 223;

49, 83, 117, 151, 185, 219, 253, 287;

55, 93, 131, 169, 207, 245, 283, 321, 359;

61, 103, 145, 187, 229, 271, 313, 355, 397, 439;

MATHEMATICA

t[n_, k_]:=4 n*k + 2n + 2k - 1; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

PROG

(Magma) [4*n*k + 2*n + 2*k - 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

(PARI) for(n=1, 10, for(k=1, n, print1(4*n*k + 2*n + 2*k - 1, ", "))) \\ G. C. Greubel, Aug 02 2017

CROSSREFS

Cf. A016921, A017305, A126980, A155151, A155156.

KEYWORD

nonn,easy,tabl

AUTHOR

Vincenzo Librandi, Aug 03 2009

STATUS

approved

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