A289265 -id:A289265 - cp's OEIS frontend

A289260 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=8/5.

1, 3, 5, 9, 17, 30, 52, 90, 154, 262, 446, 758, 1286, 2182, 3702, 6278, 10646, 18054, 30614, 51910, 88022, 149254, 253078, 429126, 727638, 1233798, 2092054, 3547334, 6014934, 10199046, 17293718, 29323590, 49721686, 84309126, 142956310, 242399686, 411017942

Offset: 0

Clark Kimberling, Jul 14 2017

Conjecture: the sequence is strictly increasing.

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

Cf. A078140 (includes guide to related sequences), A289265.

Cf. A279780.

r = 8/5;
u = 1000; (* # initial terms from given series *)
v = 100;   (* # coefficients in reciprocal series *)
CoefficientList[Series[1/Sum[Floor[r*(k + 1)] (-x)^k, {k, 0, u}], {x, 0, v}], x]
LinearRecurrence[{2,-1,2,-2},{1,3,5,9,17,30,52},40] (* Harvey P. Dale, Oct 13 2023 *)

Vec((1 + x)^2*(1 - x + x^2 - x^3 + x^4) / ((1 - x)*(1 - x - 2*x^3)) + O(x^50)) \\ Colin Barker, Jul 20 2017

G.f.: 1/(Sum_{k>=0} [(k+1)*r](-x)^k), where r = 8/5 and [ ] = floor.
From Colin Barker, Jul 14 2017: (Start)
G.f.: (1 + x)^2*(1 - x + x^2 - x^3 + x^4) / ((1 - x)*(1 - x - 2*x^3)).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 2*a(n-4) for n>3.
(End)
a(n) = abs(A279780(n)). - Alois P. Heinz, Jul 15 2017

A338089 Minimal number of moves for the cyclic variant of Hanoi's tower for 4 pegs and n disks, with the final peg three steps away.

Original entry on oeis.org

3, 10, 21, 40, 75, 134, 233, 400, 683, 1166, 1981, 3364, 5711, 9690, 16433, 27872, 47267, 80150, 135909, 230460, 390775

Offset: 1

Paul Zimmermann, Oct 09 2020

			For n=2, assume the two disks are on North initially, first move the smallest one to South in 2 moves, then the largest one to East in 1 move, the smallest one back to North in 2 moves, the largest one to West in 2 moves, and finally the smallest one to West in 3 moves, with a total of 10 moves. Each disk has a number of moves which is 3 mod 4, thus a(n) == 3*n (mod 4).

Martin Ehrenstein, (C++) Program for A338024 (computes terms of this sequence, too)

Cf. A292764, A338024.

Conjecture: a(n) = a(n-1) + a(n-2) + a(n-3) - 2*a(n-5) for n > 9 (the same recurrence as conjectured in A292764 and A338024). - Pontus von Brömssen, Oct 12 2020

a(n) ~ k*r^n, k = (725 + (310451786 - 3203949*sqrt(87))^(1/3) + (310451786 + 3203949*sqrt(87))^(1/3))/348, r=constant of A289265 (closed-form by Amiram Eldar via von Brömssen conjecture). - Bill McEachen, Aug 19 2025

a(17)-a(21) from Martin Ehrenstein, Oct 26 2020

A272031 Decimal expansion of the Hausdorff dimension of the Heighway-Harter dragon curve boundary.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 18 2016

The value for 'twindragon' is the same.

			1.5236270862024921062776839359542166272849363834011934781386909094...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Angel Chang and Tianrong Zhang, On the Fractal Structure of the Boundary of Dragon Curve, Journal of Recreational Mathematics, volume 30, number 1, 1999-2000, pages 9-22. See also the pdf version.
Eric Weisstein's World of Mathematics, Dragon curve.
Wikipedia, Dragon curve.
Wikipedia, List of fractals by Hausdorff dimension.

Cf. A014577, A191689 (Levy dragon), A327620 (tame twin-dragon).

RealDigits[Log2[(1 + (73+6*Sqrt[87])^(1/3) + (73-6*Sqrt[87])^(1/3))/3], 10, 100][[1]] (* Amiram Eldar, May 18 2021 *)

log((1+(73+6*sqrt(87))^(1/3)+(73-6*sqrt(87))^(1/3))/3)/log(2)

Equals log_2((1+(73+6*sqrt(87))^(1/3)+(73-6*sqrt(87))^(1/3))/3).
From Kevin Ryde, Dec 06 2019: (Start)
Equals 2*log(A289265)/log(2) [Chang and Zhang, equation 9].
Equals log(A289265)/log(sqrt(2)). (End)

A317494 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 8, 4, 1, 10, 12, 1, 12, 24, 1, 14, 40, 8, 1, 16, 60, 32, 1, 18, 84, 80, 1, 20, 112, 160, 16, 1, 22, 144, 280, 80, 1, 24, 180, 448, 240, 1, 26, 220, 672, 560, 32, 1, 28, 264, 960, 1120, 192, 1, 30, 312, 1320, 2016, 672, 1, 32, 364, 1760, 3360, 1792, 64

Offset: 0

Zagros Lalo, Jul 30 2018

The numbers in rows of the triangle are along a "second layer" of skew diagonals pointing top-right in center-justified triangle given in A013609 ((1+2*x)^n) and along a "second layer" of skew diagonals pointing top-left in center-justified triangle given in A038207 ((2+x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (1+2*x)^n and (2+x)^n are given in A128099 and A207538 respectively.)
The coefficients in the expansion of 1/(1-x-2*x^3) are given by the sequence generated by the row sums.
The row sums give A003229.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.695620769559862... (see A289265), when n approaches infinity.

			Triangle begins:
  1;
  1;
  1;
  1,  2;
  1,  4;
  1,  6;
  1,  8,   4;
  1, 10,  12;
  1, 12,  24;
  1, 14,  40,    8;
  1, 16,  60,   32;
  1, 18,  84,   80;
  1, 20, 112,  160,   16;
  1, 22, 144,  280,   80;
  1, 24, 180,  448,  240;
  1, 26, 220,  672,  560,   32;
  1, 28, 264,  960, 1120,  192;
  1, 30, 312, 1320, 2016,  672;
  1, 32, 364, 1760, 3360, 1792, 64;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 358, 359

Row sums give A003229.

Cf. A013609, A038207, A289265, A317495, A128099, A207538.

Flat(List([0..20],n->List([0..Int(n/3)],k->2^k/(Factorial(n-3*k)*Factorial(k))*Factorial(n-2*k)))); # Muniru A Asiru, Jul 31 2018

t[n_, k_] := t[n, k] = 2^k/((n - 3 k)! k!) (n - 2 k)!; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]} ] // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 18}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A356035 Decimal expansion of the real root of x^3 - 2*x^2 - 1.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Aug 18 2022

This is the minimum number having the property that there are uncountably many permutation classes with the growth rate equal to that number. [Vatter] - Andrey Zabolotskiy, Dec 04 2024

			2.2055694304005903117020286177838234263771089195976994404705522035518347903...

Vincent Vatter, Small permutation classes, Proc. London Math. Soc. (3), 103 (2011), 879-921; arXiv:0712.4006 [math.CO], 2007-2016.
Wikipedia, Supersilver ratio.
Index entries for algebraic numbers, degree 3

Cf. A137421, A272874, A289265, A376121.

First[RealDigits[N[Root[#1^3-2#1^2-1 &, 1, 0], 78]]] (* Stefano Spezia, Aug 19 2022 *)

solve(x=2, 3, x^3 - 2*x^2 - 1) \\ Michel Marcus, Aug 19 2022

polrootsreal(x^3 - 2*x^2 - 1)[1] \\ Charles R Greathouse IV, Dec 04 2024

Equals ((172 + 12*sqrt(177))^(1/3)+16/(172 + 12*sqrt(177))^(1/3) + 4)/6.
Equals ((172 + 12*sqrt(177))^(1/3) + (172 - 12*sqrt(177))^(1/3)  + 4)/6.
Equals (((1/2)*(43 + 3*sqrt(3*59)))^(1/3) + ((1/2)*(43 - 3*sqrt(3*59)))^(1/3)  + 2)/3.
Equals 2*(1 + 2*cosh(log((43 + 3*sqrt(177))/16)/3))/3. - Vaclav Kotesovec, Aug 19 2022
Equals y + 2/3 where y = 1.538902... is the real root of y^3 - (4/3)*y - 43/27.
Equals 1 + A137421. - R. J. Mathar, Sep 23 2022
Equals 1/A272874. - Hugo Pfoertner, Sep 11 2024

A356030 Decimal expansion of the real root of x^3 - x - 2.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Aug 19 2022

			1.5213797068045675696040808322544385144283898284279039090904980154281564...

Cf. A289265, A356031.

RealDigits[x /. FindRoot[x^3 - x - 2, {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Jun 18 2023 *)

solve(x=1, 2, x^3 - x - 2) \\ Michel Marcus, Aug 19 2022

Equals ((27 + 3*sqrt(78))^(1/3) + 3/(27 + 3*sqrt(78))^(1/3))/3.

Equals (1 + sqrt(78)/9)^(1/3) + (1 - sqrt(78)/9)^(1/3).

cp's OEIS Frontend

A289260 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=8/5.

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A338089 Minimal number of moves for the cyclic variant of Hanoi's tower for 4 pegs and n disks, with the final peg three steps away.

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A272031 Decimal expansion of the Hausdorff dimension of the Heighway-Harter dragon curve boundary.

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A317494 Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 2 * T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

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A356035 Decimal expansion of the real root of x^3 - 2*x^2 - 1.

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A356030 Decimal expansion of the real root of x^3 - x - 2.

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