A078140 -id:A078140

A281112

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

+20
2

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

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..86.

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

EXAMPLE

c(n+1)/c(n)) -> 1.688924110769165206686359... (conjectured)

MATHEMATICA

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 *)

CROSSREFS

Cf. A078140, A001622.

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Feb 10 2017

STATUS

approved

A289265

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

+10
6

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

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

OFFSET

1,2

REFERENCES

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2.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

LINKS

Table of n, a(n) for n=1..86.

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.

FORMULA

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

EXAMPLE

1.6956207695598620574163671001175353426181793882085077...

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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).

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Jul 14 2017

STATUS

approved

A288235

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

+10
4

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

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

OFFSET

0,2

COMMENTS

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

Conjecture: the sequence is strictly increasing.

LINKS

Table of n, a(n) for n=0..35.

FORMULA

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

MATHEMATICA

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]

CROSSREFS

Cf. A078140 (includes guide to related sequences).

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 11 2017

STATUS

approved

A288236

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

+10
3

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

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

OFFSET

0,2

COMMENTS

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

Conjecture: the sequence is strictly increasing.

LINKS

Table of n, a(n) for n=0..35.

FORMULA

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

MATHEMATICA

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]

CROSSREFS

Cf. A078140 (includes guide to related sequences).

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 11 2017

STATUS

approved

A288237

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

+10
3

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

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

OFFSET

0,2

COMMENTS

Conjecture: the sequence is strictly increasing.

LINKS

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

FORMULA

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

MAPLE

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

MATHEMATICA

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]

CROSSREFS

Cf. A078140 (includes guide to related sequences).

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 11 2017

STATUS

approved

A289005

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

+10
3

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

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

1.75481735140979687233000...

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Jul 14 2017

STATUS

approved

A289032

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

+10
3

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

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

1.754817351409731128793306653817678778822...

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Jul 14 2017

STATUS

approved

A289260

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

+10
3

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

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

OFFSET

0,2

COMMENTS

Conjecture: the sequence is strictly increasing.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

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

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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

Cf. A279780.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 14 2017

STATUS

approved

A289917

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

+10
3

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

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..86.

FORMULA

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

EXAMPLE

1.722083805739042245027069212153831462070116555...

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Jul 18 2017

STATUS

approved

A289921

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

+10
3

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

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

OFFSET

0,2

COMMENTS

Conjecture: all the terms are nonnegative.

LINKS

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).

FORMULA

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

MATHEMATICA

z = 2000; r = 9/10;

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

x];

PROG

(PARI) 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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jul 18 2017

STATUS

approved