A203175 -id:A203175 - cp's OEIS frontend

A203181 T(n,k) is the number of n X k 0..2 arrays with every 1 immediately preceded by 0 to the left or above, no 0 immediately preceded by a 0, and every 2 immediately preceded by 0 1 to the left or above.

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 4, 7, 4, 3, 4, 6, 17, 17, 6, 4, 5, 10, 41, 59, 41, 10, 5, 7, 18, 97, 205, 205, 97, 18, 7, 9, 30, 235, 724, 952, 724, 235, 30, 9, 12, 50, 607, 2466, 4654, 4654, 2466, 607, 50, 12, 16, 86, 1415, 8948, 23083, 32411, 23083, 8948, 1415, 86, 16, 21, 146

Offset: 1

R. H. Hardin, Dec 30 2011

Table starts
.1..1...2....2......3.......4........5.........7..........9..........12
.1..1...2....4......6......10.......18........30.........50..........86
.2..2...7...17.....41......97......235.......607.......1415........3486
.2..4..17...59....205.....724.....2466......8948......30945......108083
.3..6..41..205....952....4654....23083....115377.....551208.....2757161
.4.10..97..724...4654...32411...223567...1625772...10889470....76035931
.5.18.235.2466..23083..223567..2208945..22411843..216858412..2141041521
.7.30.607.8948.115377.1625772.22411843.323885934.4389100997.61921804090

			Some solutions for n=5 k=3
..0..1..0....0..1..2....0..1..0....0..1..0....0..1..2....0..1..0....0..1..2
..1..0..1....1..0..1....1..0..1....1..0..1....1..0..1....1..0..1....1..0..1
..2..1..2....2..1..0....2..1..2....0..1..0....0..1..2....0..1..2....0..1..0
..0..1..2....0..1..2....0..1..0....1..2..1....1..2..0....1..2..0....1..2..1
..1..0..1....1..0..1....1..0..1....2..0..1....0..1..1....0..1..1....2..0..1

R. H. Hardin, Table of n, a(n) for n = 1..364

Column 1 is A000931(n+5). Column 2 is A203175.

A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

Original entry on oeis.org

1, 6, 9, 5, 6, 2, 0, 7, 6, 9, 5, 5, 9, 8, 6, 2, 0, 5, 7, 4, 1, 6, 3, 6, 7, 1, 0, 0, 1, 1, 7, 5, 3, 5, 3, 4, 2, 6, 1, 8, 1, 7, 9, 3, 8, 8, 2, 0, 8, 5, 0, 7, 7, 3, 0, 2, 2, 1, 8, 7, 0, 7, 2, 8, 4, 4, 5, 2, 4, 4, 5, 3, 4, 5, 4, 0, 8, 0, 0, 7, 2, 2, 1, 3, 9, 9

Offset: 1

Clark Kimberling, Jul 14 2017

			1.6956207695598620574163671001175353426181793882085077...

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2. See links in A003229.

Angel Chang and Tianrong Zhang, The Fractal Geometry of the Boundary of Dragon Curves, Journal of Recreational Mathematics, volume 30, number 1, 1999-2000, pages 9-22.
Index entries for algebraic numbers, degree 3.

Sequences growing as this power: A003229, A003476, A003479, A052537, A077949, A144181, A164395, A164399, A164410, A164414, A164471, A203175, A227036, A289260, A292764.

Cf. A078140 (includes guide to constants similar to A289260).

z = 2000; r = 8/5;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A289260 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289265 *)

solve(x=1, 2, x^3 - x^2 - 2) \\ Michel Marcus, Oct 26 2019

r = D^(1/3) + (1/9)*D^(-1/3) + 1/3 where D = 28/27 + (1/9)*sqrt(29*3) [Chang and Zhang] from the usual cubic solution formula. Or similarly r = (1/3)*(1 + C + 1/C) where C = (28 + sqrt(29*27))^(1/3). - Kevin Ryde, Oct 25 2019

A227036 Expansion of 2(1+x^2)/((1-x)(1-x-2*x^3)).

Original entry on oeis.org

2, 4, 8, 16, 28, 48, 84, 144, 244, 416, 708, 1200, 2036, 3456, 5860, 9936, 16852, 28576, 48452, 82160, 139316, 236224, 400548, 679184, 1151636, 1952736, 3311108, 5614384, 9519860, 16142080, 27370852, 46410576, 78694740, 133436448, 226257604, 383647088, 650519988, 1103035200, 1870329380

Offset: 0

Roland Kneer, Jun 28 2013

Conjecture: The perimeter of the n-th iteration of the Harter-Heighway dragon is a(n) segments or a(n)/2^(n/2) base units.
a(n) = 2^(n+1)-4*A003230(n-4) (two times the number of segments, minus four times the number of squares)
The first differences 2, 2, 4, 8, 12, 20,.. are twice the (empirical) A203175. - R. J. Mathar, Jul 02 2013

			For the 4th iteration, take two 3rd iteration dragons (2*16); put together, they will make one square, so subtract the inner perimeter 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Kevin Ryde, Iterations of the Dragon Curve, see index "B" and "R".
Helena Verrill, On the Boundary of the Harter-Heighway dragon curve, arXiv:2407.17326 [math.CO], 2024.
Wikipedia, Dragon curve
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-2).

Cf. A014577.

LinearRecurrence[{2, -1, 2, -2}, {2, 4, 8, 16}, 40] (* T. D. Noe, Jul 02 2013 *)
CoefficientList[Series[2 (1 + x^2) / ((1 - x) (1 - x - 2 x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 17 2013 *)

Vec(2*(1+x^2)/((1-x)*(1-x-2*x^3))+O(x^66)) \\ Joerg Arndt, Jul 01 2013

cp's OEIS Frontend

A203181 T(n,k) is the number of n X k 0..2 arrays with every 1 immediately preceded by 0 to the left or above, no 0 immediately preceded by a 0, and every 2 immediately preceded by 0 1 to the left or above.

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A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

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A227036 Expansion of 2(1+x^2)/((1-x)(1-x-2*x^3)).

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cp's OEIS Frontend

A203181 T(n,k) is the number of n X k 0..2 arrays with every 1 immediately preceded by 0 to the left or above, no 0 immediately preceded by a 0, and every 2 immediately preceded by 0 1 to the left or above.

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Author

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Crossrefs

A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A227036 Expansion of 2*(1+x^2)/((1-x)*(1-x-2*x^3)).

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Author

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Mathematica

PARI

A227036 Expansion of 2(1+x^2)/((1-x)(1-x-2*x^3)).