A078140 -id:A078140 - cp's OEIS frontend

A281112 Decimal expansion of (conjectured) limit c(n+1)/c(n), where c = A078140.

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

Offset: 1

Clark Kimberling, Feb 10 2017

			Limit_{n->oo} c(n+1)/c(n) = 1.688924110769165206686359... (conjectured)

Clark Kimberling, Another question about the golden ratio and other numbers, MathOverflow, Jan 17 2017.

Cf. A078140, A001622.

z = 3880; r = GoldenRatio;
f[x_] := f[x] = Sum[Floor[r*(k + 1)] x^k, {k, 0, z}];
t = CoefficientList[Series[1/f[x], {x, 0, z}], x];(* A078140 *)
Table[N[t[[n]]/t[[n - 1]], 80], {n, 2, z, 100}]
u = N[t[[z]]/t[[z - 1]], 120]
RealDigits[Abs[u], 10][[1]] (* A281112 *)

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D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2.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

A288235 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=sqrt(e).

Original entry on oeis.org

1, 3, 5, 9, 17, 30, 52, 91, 160, 281, 493, 865, 1518, 2664, 4675, 8204, 14397, 25265, 44337, 77805, 136534, 239592, 420441, 737798, 1294700, 2271961, 3986877, 6996242, 12277127, 21544115, 37805987, 66342603, 116419152, 204294349, 358499270, 629100742

Offset: 0

Clark Kimberling, Jul 11 2017

A288236(k) = a(k) if and only if k <= 56.

Conjecture: the sequence is strictly increasing.

Cf. A078140 (includes guide to related sequences), A288236.

r = Sqrt[E];
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]

G.f.: 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = sqrt(e) and [ ] = floor.

A288236 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=-4/5+sqrt(6).

Original entry on oeis.org

1, 3, 5, 9, 17, 30, 52, 91, 160, 281, 493, 865, 1518, 2664, 4675, 8204, 14397, 25265, 44337, 77805, 136534, 239592, 420441, 737798, 1294700, 2271961, 3986877, 6996242, 12277127, 21544115, 37805987, 66342603, 116419152, 204294349, 358499270, 629100742

Offset: 0

Clark Kimberling, Jul 11 2017

A288235(k) = a(k) if and only if k <= 56.

Conjecture: the sequence is strictly increasing.

Cf. A078140 (includes guide to related sequences), A288235.

r = -4/5 + Sqrt[6];
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]

G.f.: 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = -4/5 + sqrt(6) and [ ] = floor.

A288237 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=sqrt(11/4).

Original entry on oeis.org

1, 3, 5, 9, 17, 30, 52, 91, 160, 281, 493, 865, 1518, 2664, 4675, 8204, 14397, 25265, 44337, 77806, 136540, 239611, 420488, 737905, 1294933, 2272449, 3987870, 6998224, 12281027, 21551700, 37820597, 66370521, 116472145, 204394366, 358687108, 629451995

Offset: 0

Clark Kimberling, Jul 11 2017

Conjecture: the sequence is strictly increasing.

Robert Israel, Table of n, a(n) for n = 0..4089

Cf. A078140 (includes guide to related sequences).

N:= 100: # to get a(0)..a(N)
r:= sqrt(11/4):
G:= 1/add(floor((k+1)*r)*(-x)^k,k=0..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jul 13 2017

r = Sqrt[11/4];
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]

G.f.: 1/(Sum_{k>=0} [(k+1)*r]*(-x)^k), where r = sqrt(11/4) and [ ] = floor.

A289005 Decimal expansion of the limiting ratio of consecutive terms of A288235.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2017

			1.75481735140979687233000...

Cf. A288235, A289032, A078140 (includes guide to related constants).

z = 2000; r = Sqrt[E];
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288235 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A289005 *)

A289032 Decimal expansion of the limiting ratio of consecutive terms of A288236.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 14 2017

			1.754817351409731128793306653817678778822...

Cf. A288235, A289005, A078140 (includes guide to related constants).

z = 2000; r = -4/5 + Sqrt[6];
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A288236 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]](* A289032 *)

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

Original entry on oeis.org

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

A289917 Decimal expansion of the limiting ratio of consecutive terms of A289916.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 18 2017

			1.722083805739042245027069212153831462070116555...

Cf. A078140 (includes guide to related constants), A289916.

z = 2000; r = 9/7;
u = CoefficientList[Series[1/Sum[Round[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];  (* A289916  *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289917 *)

Equals (1 + sqrt(13) + sqrt(2*sqrt(13) - 2))/4. - Vaclav Kotesovec, Aug 27 2021

Largest real root of x^4 - x^3 - x^2 - x + 1. - Linas Vepstas, Feb 06 2024

A289921 Coefficients of 1/([1+r] - [1+2r]x + [1+3r]x^2 - ...), where [ ] = floor and r = 9/10.

Original entry on oeis.org

1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 2, 7, 9, 5, 1, 0, 0, 0, 0, 0, 4, 16, 25, 19, 7, 1, 0, 0, 0, 0, 8, 36, 66, 63, 33, 9, 1, 0, 0, 0, 16, 80, 168, 192, 129, 51, 11, 1, 0, 0, 32, 176, 416, 552, 450, 231, 73, 13, 1, 0, 64, 384, 1008

Offset: 0

Clark Kimberling, Jul 18 2017

Conjecture: all the terms are nonnegative.

Ray Chandler, Table of n, a(n) for n = 0..10000
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1, -1, 1, -1, 1, -1, 1, -1, 1, 1).

Cf. A078140 (includes guide to related sequences), A289922, A289923.

z = 2000; r = 9/10;
CoefficientList[Series[1/Sum[Floor[1 + (k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}],
  x];

Vec( (1 - x)*(1 + x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4) / (1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 - x^10) + O(x^100)) \\ Colin Barker, Jul 21 2017

G.f.: 1/([1+r] - [1+2r]x + [1+3r]x^2 - ...), where [ ] = floor and r = 9/10.

G.f.: (1 - x)*(1 + x)^2*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4) / (1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 - x^10). - Colin Barker, Jul 20 2017

cp's OEIS Frontend

A281112 Decimal expansion of (conjectured) limit c(n+1)/c(n), where c = A078140.

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

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A288235 Coefficients in the expansion of 1/([r]-[2*r]*x+[3*r]*x^2-...); [ ]=floor, r=sqrt(e).

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A288236 Coefficients in the expansion of 1/([r]-[2*r]*x+[3*r]*x^2-...); [ ]=floor, r=-4/5+sqrt(6).

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A288237 Coefficients in the expansion of 1/([r]-[2*r]*x+[3*r]*x^2-...); [ ]=floor, r=sqrt(11/4).

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A289005 Decimal expansion of the limiting ratio of consecutive terms of A288235.

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A289032 Decimal expansion of the limiting ratio of consecutive terms of A288236.

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

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A289917 Decimal expansion of the limiting ratio of consecutive terms of A289916.

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A288235 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=sqrt(e).

A288236 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=-4/5+sqrt(6).

A288237 Coefficients in the expansion of 1/([r]-[2r]x+[3r]x^2-...); [ ]=floor, r=sqrt(11/4).